PERIMETER AND AREA OF GEOMETRIC FIGURES

PERIMETER AND AREA OF GEOMETRIC

FIGURES

STUDY NOTES

Geometry

Geometry is used in our every day when we decorate, design, and arrange

objects in a space.

The most commonly used aspects of geometry are:

Perimeter and circumference, when dealing with the length of the edges of

an object.

Area, when dealing with the amount of surface an object covers.

Volume, when dealing with how much an object can hold.

1.2.1: Perimeter and Area of Geometric Figures

6.2.1 Learning Objectives

Define polygon

Find the perimeter of a polygon

Find the areas of common polygons

Polygons

We can make use of conversion skills with denominate numbers to make

measurements of geometric figures such as rectangles, triangles, and

circles. To make these measurements we need to be familiar with several

definitions.

Definition: Polygon

A polygon is a closed plane (flat) figure whose sides are line segments

(portions of straight lines).

Polygons

Not polygons

Perimeter

Definition: Perimeter

The perimeter of a polygon is the distance around the polygon.

To find the perimeter of a polygon, we simply add up the lengths of all the

sides.

Example 1

Find the perimeter of the rectangle.

Solution

2 cm + 5 cm + 2 cm + 5 cm

14 cm

Example 2

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Solution

4.2 mm

4.3 mm

5.4 mm

+ 9.2 mm

Example 3

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

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Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

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Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

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Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

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Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

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Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

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Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

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1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

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Find the area for the following figures below: a.

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4. Find the area of the shaded region.

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1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

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Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

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Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

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Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

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Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

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A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

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1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

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Find the area and perimeter of the following composite figures a.

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1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

(area of base) ⋅ (height)

The volume of a cone istimes

times the square of the radius times the height.

Finding Volumes of Some Common Geometric Objects

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Example 1

Find the volume of the rectangular solid.

Solution

9 in ⋅ 10 in. ⋅ 3 in

The volume of this rectangular solid is 270 cu in.

Example 2

Find the approximate volume of the sphere.

Solution

VS 3 ⋅ π ⋅ r3

( 3 ) ⋅ (3.14) ⋅ (6 cm)3

( ) ⋅ (3.14) ⋅ (216 cu cm)

904.32 cu cm

The approximate volume of this sphere is 904.32 cu cm, which is often

written as 904.32 cm3 .

Example 3

Find the approximate volume of the cylinder.

Solution

VCyl π ⋅ r2 ⋅ h

(3.14) ⋅ (4.9 ft)2 ⋅ (7.8 ft)

(3.14) ⋅ (24.01 sq ft) ⋅ (7.8 ft)

(3.14) ⋅ (187.278 cu ft)

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The volume of this cylinder is approximately 588.05292 cu ft. The volume is

approximate because we approximated .

Example 4

Find the approximate volume of the cone. Round to two decimal places.

Solution

c⋅ π ⋅ 2 ⋅ h

( 3 ) ⋅ (3.14) ⋅ (2 mm)2 ⋅ (5 mm)

( ) ⋅ (3.14) ⋅ (4 sq mm) ⋅ (5 mm)

( ) ⋅ (3.14) ⋅ (20 cu mm)

20.93 cu mm

The volume of this cone is approximately 20.93 cu mm. The volume is

approximate because we approximated .

Try It Now 1

Find the volume of the rectangular solid.

Answer

21 cu in

Try It Now 2

Find the volume of the sphere. Use the key on your calculator to find the

approximate volume.

Answer

904.32 cu ft

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Try It Now 3

Find the volume of the cylinder. Use the key on your calculator to find the

approximate volume.

Answer

157 cu m

Try It Now 4

Find the volume of the cone. Use the key on your calculator to find the

approximate volume.

Answer

0.00942 cu in

Finding the Volume of Composite Figures

Similar to finding the area of composite figures, we will first determine the

separate shapes that the figure is made of. Then use the formulas for the

volumes of the individual shapes and add them together.

Example 5

Find the approximate volume of the composite figure. Use the key on your

calculator to find the approximate volume if necessary. Round to two

decimal places.

Solution

This shape is made up of a cone and a cylinder. We can tell from the base of

the cylinder, that the diameter of the cone is 3.0 ft.

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Vcone + Vcylinder

1⋅ π ⋅ 2 ⋅ h + π ⋅ 2 ⋅ h

( ) ⋅ π ⋅ (1.5 ft)2 ⋅ (3.0 ft) + π ⋅ 1.5 ft2 ⋅ 8.1 ft.

( )⋅π⋅( 2 ) ⋅ (3.0 ft) + π ⋅2 ⋅ 8.1 ft.

( 3 ) ⋅ π ⋅ (6.75 cu ft) + π ⋅ 18.225 cu ft

7.06858 cu ft + 57.25553 cu ft

64.32 cu ft

The volume of this figure is approximately 64.32 cu ft. The volume is

approximate because we approximated .

Try It Now 6

Find the volume of the composite figure.

Answer

This shape is composed of a hemisphere and a cylinder.

cu cm

≈ 870

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1.2.3.1: Exercises

Find the approximate volume a.

2. Find the exact volume.

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ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ

Ā ᜀ Ā 倀Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ

Ā ᜀ Ā ᜀ Sketch a rectangular prism.

ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ

Ā ᜀ Ā 倀Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ

Ā ᜀ Ā ᜀ How much soil must be removed from a lot to create

a basement for a new house?

5. Two boxes in the shape of rectangular prisms are stacked one on top of

the other. What is the combined volume of the boxes?

A glass bead used to make jewelry is a rectangular prism with square

ends. The stringing hole is 1.5 mm in diameter. What is the volume of the

glass bead?

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6.2.3.1: Exercises by Leah Griffith, Veronica Holbrook, Johnny Johnson &

Nancy Garcia has no license indicated.

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1.3: Percents

1.5 Learning Objectives

Find what percent one number is of another

Change fractions to percents

Rewrite decimals as percents

Find a percent of a number

Solve applications involving percents

In the 2004 vice-presidential debates, Edwards's claimed that US forces have

suffered "90% of the coalition casualties" in Iraq. Cheney disputed this,

saying that in fact Iraqi security forces and coalition allies "have taken

almost 50 percent" of the casualties[1]. Who is correct? How can we make

sense of these numbers?

Percent literally means “per 100,” or “parts per hundred.” When we write

40%, this is equivalent to the fraction 40 or the decimal

100

0.40. Notice that 80 out of 200 and 10 out of 25 are also 40%, since .

200 = 25 = 100

Example 1

243 people out of 400 state that they like dogs. What percent is this?

Solution

60.75 . This is 60.75%.

= 0.6075 = 100

Notice that the percent can be found from the equivalent decimal by moving

the decimal point two places to the right.

Example 2

Write each as a percent: a) 1 b) 0.02 c) 2.35

Solution

a) b) c)

= 0.25 = 25% 0.02 = 2% 2.35 = 235%

Percents

If we have a part that is some percent of a whole, then

part , or equivalently,

percent = part = percent ⋅ whole

To do the calculations, we write the percent as a decimal.

Notice that the equation translated to English would be "part is percent of

the whole." So direct multiplication is used.

In some cases the question will ask for the percent off the whole, which is

different.

If we have a part that is some percent off a whole, then

part = (100% − percent) ⋅ whole

Since the statement is the percent off, to find the correct value for the part,

the percent must be subtracted from 100% first.

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Example 3

The sales tax in a town is 9.4%. How much tax will you pay on a $140

purchase?

Solution

Here, $140 is the whole, and we want to find 9.4% of $140. We start by

writing the percent as a decimal by moving the decimal point two places to

the left (which is equivalent to dividing by 100). We can then compute:

in tax.

tax = 0.094(140) = $13.16

Example 4

In the news, you hear “tuition is expected to increase by 7% next year.” If

tuition this year was $1200 per quarter, what will it be next year?

Solution

The tuition next year will be the current tuition plus an additional 7%, so it

will be 107% of this year’s tuition:

$1200(1.07)

= $1284

Alternatively, we could have first calculated 7% of $1200:

$1200(0.07)

= $84

Notice this is not the expected tuition for next year (we could only wish).

Instead, this is the expected increase, so to calculate the expected tuition,

we’ll need to add this change to the previous year’s tuition:

$1200 + $84 = $1284

Try it Now 1

A TV originally priced at $799 is on sale for 30% off. There is then a 9.2%

sales tax. Find the price after including the discount and sales tax.

Answer

The sale price is

. After tax, the price is

$799(1 − 0.30) = $799(0.70) = $559.30

$559.30(1.092)

= $610.76

Example 5

The value of a car dropped from $7400 to $6800 over the last year. What

percent decrease is this?

Solution

To compute the percent change, we first need to find the dollar value

change: Often we will take the

absolute value of this amount, which is called the absolute change:

$6800 − $7400 = −$600

| − 600| = 600

Since we are computing the decrease relative to the starting value, we

compute this percent out of :

decrease. This is called a relative change. $7400

600

7400 = 0.081 = 8.1%

Absolute and Relative Change

Given two quantities,

Absolute change =

|ending quantity − starting quantity|

Relative change: absolute change

starting quantity

Absolute change has the same units as the original quantity.

Relative change gives a percent change.

The starting quantity is called the base of the percent change.

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The base of a percent is very important. For example, while Nixon was

president, it was argued that marijuana was a “gateway” drug, claiming that

80% of marijuana smokers went on to use harder drugs like cocaine. The

problem is, this isn’t true. The true claim is that 80% of harder drug users

first smoked marijuana. The difference is one of base: 80% of marijuana

smokers using hard drugs, vs. 80% of hard drug users having smoked

marijuana. These numbers are not equivalent. As it turns out, only one in

2,400 marijuana users actually go on to use harder drugs[2].

Example 6

There are about 75 QFC supermarkets in the U.S. Albertsons has about 215

stores. Compare the size of the two companies.

Solution

When we make comparisons, we must ask first whether an absolute or

relative comparison. The absolute difference is

From this, we could say “Albertsons has 140 more stores than QFC.”

However, if you wrote this in an article

215– 75 = 140

or paper, that number does not mean much. The relative difference may be

more meaningful. There are two different relative changes we could

calculate, depending on which store we use as the base:

Using QFC as the base, .

= 1.867

This tells us Albertsons is 186.7% larger than QFC.

Using Albertsons as the base, 140 .

= 0.651

This tells us QFC is 65.1% smaller than Albertsons.

Notice both of these are showing percent differences.

We could also calculate the size of Albertsons relative to QFC: 215/75, which

means Albertsons is 2.867 or 286.7% times the size of QFC. Likewise, we

could calculate the size of QFC relative to Albertsons: 75/215, which tells us

that QFC is 34.9% of the size of Albertsons.

Example 7

Suppose a stock drops in value by 60% one week, then increases in value

the next week by 75%. Is the value higher or lower than where it started?

Solution

To answer this question, suppose the value started at $100. After one week,

the value dropped by 60%:

$100 − $100(0.60) = $100 − $60 = $40

In the next week, notice that base of the percent has changed to the new

value, $40. Computing the 75% increase:

$40 + $40(0.75) = $40 + $30 = $70

In the end, the stock is still $30 lower, or $30 lower, valued than it started.

$100 = 30%

Try it Now 2

The U.S. federal debt at the end of 2001 was $5.77 trillion, and grew to $6.20

trillion by the end of 2002. At the end of 2005 it was $7.91 trillion, and grew

to $8.45 trillion by the end of 2006[3]. Calculate the absolute and relative

increase for 2001-2002 and 2005-2006. Which year saw a larger increase in

federal debt?

Answer

2001-2002: Absolute change: $0.43 trillion. Relative change: 7.45%

2005-2006: Absolute change: $0.54 trillion. Relative change: 6.83%

2005-2006 saw a larger absolute increase, but a smaller relative increase.

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Example 8

A Seattle Times article on high school graduation rates reported “The

number of schools graduating 60 percent or fewer students in four years –

sometimes referred to as “dropout factories” – decreased by 17 during that

time period. The number of kids attending schools with such low graduation

rates was cut in half.”

Is the “decrease by 17” number a useful comparison?

Considering the last sentence, can we conclude that the number of

“dropout factories” was originally 34?

Solution

This number is hard to evaluate, since we have no basis for judging

whether this is a large or small change relative to the number of such

schools. If the number of “dropout factories” dropped from 20 to 3, that

would be a very significant change, but if the number dropped from 217 to

200, that would be less of an improvement.

The last sentence provides relative change which helps put the first

sentence in perspective. We can estimate that the number of “dropout

factories” was probably previously around 34. However, it’s possible that

students simply moved schools rather than the school improving, so that

estimate might not be fully accurate.

Example 9

In the 2004 vice-presidential debates, Edwards's claimed that US forces have

suffered "90% of the coalition casualties" in Iraq. Cheney disputed this,

saying that in fact Iraqi security forces and coalition allies "have taken

almost 50 percent" of the casualties. Who is correct?

Solution

Without more information, it is hard for us to judge who is correct, but we

can easily conclude that these two percents are talking about different

things, so one does not necessarily contradict the other. Edward’s claim was

a percent with coalition forces as the base of the percent, while Cheney’s

claim was a percent with both coalition and Iraqi security forces as the base

of the percent. It turns out both statistics are in fact fairly accurate.

Try it Now 3

In the 2012 presidential elections, one candidate argued that “the

president’s plan will cut $716 billion from Medicare, leading to fewer services

for seniors,” while the other candidate rebuts that “our plan does not cut

current spending and actually expands benefits for seniors, while

implementing cost saving measures.” Are these claims in conflict, in

agreement, or not comparable because they’re talking about different

things?

Answer

Without more information, it is hard to judge these arguments. This is

compounded by the complexity of Medicare. As it turns out, the $716 billion

is not a cut in current spending, but a cut in future increases in spending,

largely reducing future growth in health care payments. In this case, at least

the numerical claims in both statements could be considered at least

partially true. Here is one source of more information if you’re interested:

http://factcheck.org/2012/08/a-campaign-full-of-mediscare/

We’ll wrap up our review of percents with a few cautions.

First, when talking about a change in quantities that are already measured in

percents, we have to be careful in how we describe the change.

Example 10

A politician’s support increases from 40% of voters to 50% of voters.

Describe the change.

Solution

We could describe this using an absolute change: . Notice that since the

original quantities were percents,

|50% − 40%| = 10%

this change also has the units of percent. In this case, it is best to describe

this as an increase of 10 percentage points.

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In contrast, we could compute the percent change: 10% increase. This is

the relative change, and we’d say the

politician’s support has increased by 25%.

40% = 0.25 = 25%

Second, pay attention to shifting base values.

Example 11

An employer asks his employee to take a temporary 10% pay cut for 6

months because the company has hit some hard times. At the end of the 6

months the employer will give the employee a 10% raise. Will the employee

make more, less, or the same amount as before the pay cut? Is that fair?

Solution

Suppose the employee makes $4500 per month. The employer is proposing

to take 10% off the current salary.

part = $4500 (1 − 0.10) = $4500 (0.90) = $4050

After 6 months pass the employer promises to give a 10% raise on the

salary. The employee will retain 100% of what they have plus 10%.

part = $4050 (1 + 0.10) = $4050 (1.10) = $4455

Since the value that the percent is based on has changed the employee

would not be restored to their original salary in this scenario.

Last, beware of averaging percents.

Example 12

A basketball player scores on 40% of 2-point field goal attempts, and on 30%

of 3-point of field goal attempts. Find the player’s overall field goal

percentage.

Solution

It is very tempting to average these values, and claim the overall average is

35%, but this is likely not correct, since most players make many more 2-

point attempts than 3-point attempts. We don’t actually have enough

information to answer the question. Suppose the player attempted 200 2-

point field goals and 100 3-point field goals. Then they made 200(0.40) = 80

point shots and 3-point shots. Overall, they made 110 shots out of 300, for

a 110 overall

100(0.30) = 30 300 = 0.367 = 36.7%

field goal percentage.

www.factcheck.org/cheney_edwards_mangle_facts.html

http://tvtropes.org/pmwiki/pmwiki.php/Main/LiesDamnedLiesAndStatistics

www.whitehouse.gov/sites/defa...s/hist07z1.xls

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1.3.5 146712

1.3.1: Exercises

Section 1.5.1

Write each percent as a fraction.

87%

40%

110%

Write as a percent.

5858

4545

Write each percent as a decimal.

27%

8.328%

125%

0.5%

Change each decimal to a percent.

0.34

2.5

0.0045

50 is 40% of what number?

A shirt that usually sells for $25 is on sale for $21. What is the percent

decrease in the price of the shirt?

A car salesman has a commission rate of 12% on the cars sold. If the

commission on one sale was $1836, what was the purchase price of the car?

You purchase an item originally priced at $240 that is discounted 30%.

What is the sale price of the item?

You pay sales tax of 9.5% on your purchase. What is the total cost for

your purchase with the discount and the sales tax?

The sales clerk accidentally added the sales tax based on the original

price of your purchase before giving you a 30% discount. Would this error

cause the item to cost you more or less or the same as what you figured in

part b)?

A store advertises a 40% off clearance sale. When you arrived at the

store, you received a coupon for an additional 10% off. The coupon is applied

after the 40% discount.

What is the total discount if the item is normally priced at $30?

What is the total discount percent?

Why do you think the store uses two discount percents instead of one?

Explain.

Consider the following.

A parenting blog states that 12-year-old girls typically weigh 200% of

what 6-year-old girls weigh? Does that mean a girl’s weight doubles between

ages 6 and 12?

Could it be true that an 8-year-old girl weighs 30% more than a 6-year-

old girl?

Out of 230 racers who started the marathon, 212 completed the race,

14 gave up, and 4 were disqualified. What percentage did not complete the

marathon?

Patrick left an $8 tip on a $50 restaurant bill. What percent tip is that?

Ireland has a 23% VAT (value-added tax, similar to a sales tax). How

much will the VAT be on a purchase of a €250 item?

Employees in 2012 paid 4.2% of their gross wages towards social

security (FICA tax), while employers paid another 6.2%.

How much will someone earning $45,000 a year pay towards social

security out of their gross wages?

What is the total amount contributed towards the worker's social

security?

A project on Kickstarter.com was aiming to raise $15,000 for a

precision coffee press. They ended up with 714 supporters, raising 557% of

their goal. How much did they raise?

Another project on Kickstarter for an iPad stylus raised 1,253% of their

goal, raising a total of $313,490 from 7,511 supporters. What was their

original goal?

The population of a town increased from 3,250 in 2008 to 4,300 in

2010. Find the absolute and relative (percent) increase.

1.3.1.1 146713

The number of CDs sold in 2010 was 114 million, down from 147

million the previous year[1]. Find the absolute and relative (percent)

decrease.

A company wants to decrease their energy use by at least 15%. If their

next bill is $1,700 a month, were they successful? Why or why not?

A store is hoping an advertising campaign will increase their number of

customers by at least 30%. They currently have about 80 customers a day.

How many customers will they have if their campaign is successful?

If they increase to 120 customers a day, were they successful? Why or

why not?

An article reports “attendance dropped 6% this year, to 300.” What

was the attendance before the drop?

An article reports “sales have grown by 30% this year, to $200

million.” What were sales before the growth?

The Walden University had 47,456 students in 2010, while Kaplan

University had 77,966 students. Complete the following statements:

Kaplan’s enrollment was ___% larger than Walden’s.

Walden’s enrollment was ___% smaller than Kaplan’s.

Walden’s enrollment was ___% of Kaplan’s.

In the 2012 Olympics, Usain Bolt ran the 100m dash in 9.63 seconds.

Jim Hines won the 1968 Olympic gold with a time of 9.95 seconds.

Bolt’s time was ___% faster than Hines’.

Hines’ time was ___% slower than Bolt’s.

Hines’ time was ___% of Bolt’s.

A store has clearance items that have been marked down by 60%.

They are having a sale, advertising an additional 30% off clearance items.

What percent of the original price do you end up paying?

Which is better: having a stock that goes up 30% on Monday than

drops 30% on Tuesday, or a stock that drops 30% on Monday and goes up

30% on Tuesday? In each case, what is the net percent gain or loss?

Are these two claims equivalent, in conflict, or not comparable because

they’re talking about different things?

“16.3% of Americans are without health insurance”[2]

“only 55.9% of adults receive employer provided health insurance”[3]

Are these two claims equivalent, in conflict, or not comparable because

they’re talking about different things?

“We mark up the wholesale price by 33% to come up with the retail

price”

“The store has a 25% profit margin”

Are these two claims equivalent, in conflict, or not comparable because

they’re talking about different things?

“Every year since 1950, the number of American children gunned

down has doubled.”

“The number of child gunshot deaths has doubled from 1950 to 1994.”

Are these two claims equivalent, in conflict, or not comparable because

they’re talking about different things?[4]

“75 percent of the federal health care law’s taxes would be paid by

those earning less than $120,000 a year”

“75 percent of those who would pay the penalty [health care law’s

taxes] for not having insurance in 2016 would earn under $120,000”

Are these two claims equivalent, in conflict, or not comparable because

they’re talking about different things?

“The school levy is only a 0.1% increase of the property tax rate.”

“This new levy is a 12% tax hike, raising our total rate to $9.33 per

$1000 of value.”

A high school currently has a 30% dropout rate. They’ve been tasked

to decrease that rate by 20%. Find the equivalent percentage point drop.

A politician’s support grew from 42% by 3 percentage points to 45%.

What percent (relative) change is this?

Marcy has a 70% average in her class going into the final exam. She

says "I need to get a 100% on this final so I can raise my score to 85%." Is

she correct?

In a survey of households in the United States 9.4% of the households

had 2 dogs. If there are about 117 million households in the U.S., about how

many households have 2 dogs?

A typical 180-lb man is 111 lbs of water and 27 pounds of fat. What

percent of the weight is water? What percent of the weight is fat?

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Between the last two censuses, a town’s population grew from 3200 to

3650. What is the percent increase in the population?

In 1901 life expectancy at birth was 49 years. At the end of the century

life expectancy at birth was 77 years. What is the increase in life expectancy

during the 20th century? Complete this sentence: Life expectancy in 2000

was ______ percent of life expectancy in 1901.

If you bought a stock for $250 and later sold it for $1250, by what

percent did your stock increase in value? What percent of the purchase price

was the sales price?

At work, you get a 2% cost-of-living adjustment (COLA). True of False:

Over a 5-year period, your pay will increase by 10%. If this is false, by what

percent will your pay increase over the 5-year period?

All merchandise at a store is on sale at 30% off. You have a coupon

that gives you 10% off the purchase price. What will you pay for a jacket

originally priced at $80? What percent off of the original price is your final

price?

Your first three exam scores in a class are 74%, 89%, and 83%. What

percent do you need to receive on the last exam to have an 85% overall in

the class?

What is the markup percent for an item that has a wholesale price of

$1.13 and a retail price of $1.69?

If you buy a refrigerator priced at $1325 and the sales tax rate is 6.5%,

how much sales tax would you pay on the purchase? What would the total

price of the refrigerator be?

Find the sales tax rate in a region where you paid $62.65 tax on a big

screen TV priced at $895.

Why is this conclusion incorrect? “In a survey of a class, 70% want

pizza for their class party and 20% want cake. Therefore, 90% want pizza or

cake for their party.”

Why is this reasoning incorrect? “You received a score of 80% on five

tests in a class and 90% on one test. Therefore, your average test score is

85% for the six tests.”

Suppose you have one quart of water/juice mix that is 50% juice, and

you add 2 quarts of juice. What percent juice is the final mix?

During the 2004-2005 NBA, Kobe Bryant attempted 387 3-point shots

and made 131 of them. Express his performance as a fraction, a decimal and

a percent.

The following table describes the students enrolled at a small two-year

college.

class full-time part-time

freshman 262 116

sophomore 214 239

What percent of the freshmen are full-time students?

What percent of the students are enrolled part-time?

What percent of the part-time students are sophomores?

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5 . We can solve this by

1.3.2: Proportions and Rates

1.5.1 Learning Objectives

Compare 2 quantities as a rate or ratio

Solve a proportion

Solve applications of proportions

Solve applications using dimensional analysis

If you wanted to power the city of Seattle using wind power, how many

windmills would you need to install? Questions like these can be answered

using rates and proportions.

Rates

A ratio compares two quantities with the same units by division.

A rate is the ratio (fraction) of two quantities with different units.

A unit rate is a rate with a numerator or a denominator of one.

Example 1

Your car can travel 300 miles on a tank of 15 gallons. Express this as a rate

and as a unit rate.

Solution

Expressed as a rate, 300 miles . We can divide to find a unit rate: 20

miles , which we could also write asmiles , or

15 gallons 1 gallon 20 gallon

just 20 miles per gallon.

Proportion Equation

A proportion equation is an equation showing the equivalence of two rates or

ratios.

Example 2

Solve the proportion for the unknown value .

Solution

This proportion is asking us to find a fraction with denominator 6 that is

equivalent to the fraction

multiplying both sides of the equation by 6, giving .

x = ⋅ 6 = 10

You could also note that the denominator of the right-hand ratio is twice as

large as the denominator of the left-hand ratio, so the numerator on the right

must also be twice as large as the numerator on the left.

Example 3

A map scale indicates that inch on the map corresponds with 3 real miles.

How many miles apart are two cities that are 1 2 4

inches apart on the map?

Solution

We can set up a proportion by setting equal two map inches rates, and

introducing a variable, , to represent the unknown quantity – the mile

distance between the cities.

1.3.2.1 146714

map inch 1

2 4 map inches

3 miles = Multiply

both sides by x and rewriting the mixed number

x miles

⋅ x = 9

Multiply both sides by 3

3 4

x = Multiply

both sides by 2 (or divide by )

x = 2 = 13 2 miles

Many proportion problems can also be solved using dimensional analysis, the

process of multiplying a quantity by rates to change the units.

Example 4

Your car can travel 300 miles on a tank of 15 gallons. How far can it travel on

40 gallons?

Solution

We could certainly answer this question using a proportion: 300 miles x

miles .

15 gallons = 40 gallons

However, we earlier found that 300 miles on 15 gallons gives a rate of 20

miles per gallon. If we multiply the given 40 gallon quantity by this rate, the

gallons unit “cancels” and we’re left with a number of miles:

gallons

40gallons ⋅gallon = ⋅gallon = 800 miles

Notice if instead we were asked “how many gallons are needed to drive 50

miles?” we could answer this question by inverting the 20 mile per gallon

rate so that the miles unit cancels and we’re left with gallons:

1 gallon 50 1 gallon 50 gallons

50miles ⋅ 20 miles = 1 ⋅ 20 = 20 = 2.5 gallons

Dimensional analysis can also be used to do unit conversions. Here are some

unit conversions for reference.

Unit Conversions

Length

1 foot (ft) = 12 inches (in)

1 mile = 5, 280 feet

1000 millimeters mm = 1 meter (m)

1000 meters (m) = 1 kilometer (km)

Weight and Mass

1 pound (lb) = 16 ounces (oz)

1000 milligrams (mg) = 1 gram (g)

1 kilogram = 2.2 pounds (on earth)

Capacity

1 yard (yd) = 3 feet (ft)

100 centimeters (cm) = 1 meter

2.54 centimeters (cm) = 1 inch

1 ton = 2000 pounds

1000 grams = 1 kilogram (kg)

1 cup = 8 fluid ounces (fl oz) ∗1 pint = 2 cups

1 quart = 2 pints = 4 cups 1 gallon = 4 quarts = 16 cups

1000 milliliters (ml) = 1 liter (L)

*Fluid ounces are a capacity measurement for liquids. 1 fluid ounce ≈ 1

ounce (weight) for water only.

1.3.2.2 146714

Example 5

A bicycle is traveling at 15 miles per hour. How many feet will it cover in 20

seconds?

Solution

To answer this question, we need to convert 20 seconds into hours. If we

know the speed of the bicycle in feet per second, this question would be

simpler. Since we don’t, we will need to do additional unit conversions. We

will need to know that 5280 ft = 1 mile. We might start by converting the 20

seconds into hours:

Now we can multiply by the 15 miles/hr

20 ⋅ 60 ⋅60 minutes = 180 hour

180 ⋅= mile

⋅5280 feet = 440 feet

We could have also done this entire calculation in one long set of products:

1 1 15

20 ⋅ 60 seconds ⋅ 60 ⋅ 1 hour ⋅ 1 mile = 440 feet

Try it Now 1

A 1000 foot spool of bare 12-gauge copper wire weighs 19.8 pounds. How

much will 18 inches of the wire weigh, in ounces?

Answer

19.8 pounds

18 inches ⋅ 12 inches ⋅1000 feet ⋅ 1 pound ≈ 0.475 ounces

Notice that with the miles per gallon example, if we double the miles driven,

we double the gas used. Likewise, with the map distance example, if the

map distance doubles, the real-life distance doubles. This is a key feature of

proportional relationships, and one we must confirm before assuming two

things are related proportionally.

Example 6

Suppose you’re tiling the floor of a 10 ft by 10 ft room, and find that 100 tiles

will be needed. How many tiles will be needed to tile the floor of a 20 ft by 20

ft room?

Solution

In this case, while the width the room has doubled, the area has quadrupled.

Since the number of tiles needed corresponds with the area of the floor, not

the width, 400 tiles will be needed. We could find this using a proportion

based on the areas of the rooms:

100 tiles

2 = 2

Other quantities just don’t scale proportionally at all.

Example 7

Suppose a small company spends $1000 on an advertising campaign, and

gains 100 new customers from it. How many new customers should they

expect if they spend $10,000?

Solution

While it is tempting to say that they will gain 1000 new customers, it is likely

that additional advertising will be less effective than the initial advertising.

For example, if the company is a hot tub store, there are likely only a fixed

number of people interested in buying a hot tub, so there might not even be

1000 people in the town who would be potential customers.

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Sometimes when working with rates, proportions, and percents, the process

can be made more challenging by the magnitude of the numbers involved.

Sometimes, large numbers are just difficult to comprehend.

Example 8

Compare the 2010 U.S. military budget of $683.7 billion to other quantities.

Here we have a very large number, about $683,700,000,000 written out. Of

course, imagining a billion dollars is very difficult, so it can help to compare it

to other quantities.

If that amount of money was used to pay the salaries of the 1.4 million

Walmart employees in the U.S., each would earn over $488,000.

There are about 300 million people in the U.S. The military budget is about

$2,200 per person.

If you were to put $683.7 billion in $100 bills, and count out 1 per second, it

would take 216 years to finish counting it.

Example 9

Compare the electricity consumption per capita in China to the rate in Japan.

To address this question, we will first need data. From the CIA[1] website we

can find the electricity consumption in 2011 for China was

4,693,000,000,000 KWH (kilowatt-hours), or 4.693 trillion KWH, while the

consumption for Japan was 859,700,000,000, or 859.7 billion KWH. To find

the rate per capita (per person), we will also need the population of the two

countries. From the World Bank[2], we can find the population of China is

1,344,130,000, or 1.344 billion, and the population of Japan is 127,817,277,

or 127.8 million.

Solution

Computing the consumption per capita for each country:

China:

Japan:

4, 693, 000, 000, 000KWH KWH per person

1, 344, 130, 000 people≈ 3491.5

859, 700, 000, 000KWH KWH per person

127, 817, 277 people ≈ 6726

While China uses more than 5 times the electricity of Japan overall, because

the population of Japan is so much smaller, it turns out Japan uses almost

twice the electricity per person compared to China.

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1.3.2.4 146714

1.3.3: Exercises

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

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1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

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1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

1.2.2.2 146708

Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

1.2.2.3 146708

Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

1.2.2.4 146708

Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

1.2.2.5 146708

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

This page titled 1.2.2: Circles is shared under a not

1.2.2.6 146708

1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

1.2.2.1.1 146709

Find the area and perimeter of the following composite figures a.

1.2.2.1.2 146709

This page titled 1.2.2.1: Exercises is shared under a not

1.2.2.1.3 146709

1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Solution

Our first observation is that three of the dimensions are missing. However,

we can determine the missing measurements using the following process.

Let A, B, and C represent the missing measurements. Visualize

A = 12m - 2m = 10m

B = 9m + 1m - 2m = 8m

C = 12m - 1m = 11m

8 m

+ 1 m

44 m

1.2.1.2 146706

Try It Now 1

Find the perimeter of this triangle.

Answer

20 ft

Try It Now 2

Find the perimeter of this polygon.

Answer

46 cm

Try It Now 3

Answer

26.8 m

Try It Now 4

1.2.1.3 146706

Answer

49.89 mi

Quite often it is necessary to multiply one denominate number by another.

To do so, we multiply the number parts together and the unit parts together.

For example,

64 in2

Sometimes the product of units has a physical meaning. In this section, we

will examine the meaning of the products

(length unit) 2 and (length unit) 3

The Meaning and Notation for Area

The product (length unit) ⋅ (length unit) = (length unit) 2 , or, square length

unit (sq length unit), can be interpreted physically as the area of a surface.

Area

The area of a surface is the number of square length units contained in the

surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be

placed precisely on some surface. (The squares may have to be cut and

rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

Area Formulas

We can determine the areas of these geometric figures using the following

formulas.

Figure

Area Formula

Statement

1.2.1.4

146706

Triangle

T = ⋅ b ⋅ h

Rectangle

R = l ⋅ w

Parallelogram

P = b ⋅ h

Trapezoid

Trap = ⋅ ( 1 + 2 ) ⋅ h

Area of a triangle is one half the base times the height.

Area of a rectangle is the length times the width.

Area of a parallelogram is base times the height.

Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

Solution

2 ⋅ b ⋅ h

20 ⋅ 5 sq ft

10 ⋅ 6 sq ft

60 sq ft

60 ft2

The area of this triangle is 60 sq ft, which is often written as 60 2 .

Example 5

Find the area of the rectangle.

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll

use the unit fraction since it has inches in

the numerator. Then,

4 ft 12 in

4 ft ⋅

⋅ 1

1.2.1.5 146706

Thus,

4 ft 2 in = 48 in + 2 in = 50 in

50 in ⋅ 8 in

400 sq in

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

Solution

63.86 sq cm

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

Solution

ATrap2 ⋅ (b1 + b2 ) ⋅ h

⋅ (14.5 mm + 20.4 mm) ⋅ (4.1 mm)

⋅ (34.9 mm) ⋅ (4.1 mm)

⋅ (143.09 sq mm)

71.545 sq mm

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

Answer

36 sq cm

1.2.1.6 146706

Try It Now 6

Find the area of the rectangle

Answer

37.503 sq mm

Try It Now 7

Answer

13.26 sq in.

Try It Now 8

Answer

367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what

shapes the composite figure is made of first. Once you have determined the

shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

1.2.1.7 146706

Solution

This figure is made up of two squares and one rectangle. There is a square

with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but

to figure out the other side we need to subtract 9 cm 2 cm 7 cm.

−

figure

l ⋅ w + l ⋅ w + l ⋅ w

1 cm ⋅ 1 cm + 2 cm ⋅

2 cm + 12 cm ⋅ 7 cm

2 + 2

2 +84

87sq cm

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

This page titled 1.2.1: Perimeter and Area of Geometric Figures is shared

under a CC BY license and was authored, remixed,

1.2.1.8 146706

1.2.1.1: Exercises

Find each perimeter. a.

Find the area for the following: a.

1.2.1.1.1 146707

Find the area for the following figures below: a.

1.2.1.1.2 146707

4. Find the area of the shaded region.

This page titled 1.2.1.1: Exercises is shared under a not

1.2.1.1.3 146707

1.2.2: Circles

6.2.2 Learning Objectives

Identify the radius and diameter of a circle.

Use or its approximate value in formulas.

Find the circumference of a circle.

Find the area of a circle.

Find the perimeter or area of composite figures involving circles/

Circumference/Diameter/Radius

Definition: Part of a Circle

The circumference of a circle is the distance around the circle.

A diameter of a circle is any line segment that passes through the center of

the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of

the circle and a point on the circle.

The radius is one half the diameter.

The Number

The symbol , read "pi," represents the nonterminating, nonrepeating decimal

number 3.14159 … . This number has been

computed to millions of decimal places without the appearance of a

repeating block of digits.

For computational purposes, is often approximated as 3.14. We will write

to denote that is approximately equal to

3.14. The symbol "≈" means "approximately equal to." Use the key on your

calculator when evaluating for the best approximation.

Formulas

To find the circumference of a circle, we need only know its diameter or

radius. We then use a formula for computing the circumference of the circle.

Definition: Formula

A formula is a rule or method for performing a task. In mathematics, a

formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but

possibly unknown, quantities.

If , and represent, respectively, the circumference, diameter, and radius

of a circle, then the following two formulas give us

C, d

directions for computing the circumference of the circle.

Circumference Formulas

1.2.2.1 146708

Example 1

Find the exact circumference of the circle.

Solution

Use the formula .

C = πd

By commutativity of multiplication,

= 7 in ⋅ π

exactly

This result is exact since has not been approximated.

Example 2

Find the approximate circumference of the circle.

Solution

Use the formula .

≈ (3.14)(6.2)

This result is approximate since has been approximated by 3.14.

Example 3

Find the approximate circumference of a circle with radius 18 inches.

Solution

Since we're given that the radius, , is 18 in, we'll use the formula.

≈ (2)(3.14)(18 in)

Example 4

Find the approximate area of the figure.

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Solution

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

5.1 cm

2.0 cm

(0.5) ⋅ (2) ⋅ (3.14) ⋅ (6.2 cm) Circumference of outer semicircle.

+ (0.5) ⋅ (2) ⋅ (3.14) ⋅ (4.2 cm)

The 0.5 appears because we want the

perimeter of only half a circle.

Perimeter 2.0 cm

5.1 cm

+13.188 cm

Try It Now 1

Find the exact circumference of the circle.

Answer

9.1π

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Try It Now 2

Find the approximate circumference of the circle.

Answer

5.652 mm

Try It Now 3

Find the approximate circumference of the circle with radius 20.1 m.

Answer

126.228 m

Try It Now 4

Find the approximate outside perimeter of

Answer

41.634 mm

Area of a Circle

Figure Area Formula Statement

Circle c = π 2 Area of a circle is times the

square of the radius.

Example 5

Find the approximate area of the circle.

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Solution

(3.14) ⋅ (16.8 ft)2

(3.14) ⋅ (282.24 sq ft)

888.23 sq ft

The area of this circle is approximately 886.23 sq ft.

Example 6

Find the approximate area of the circle.

Solution

In this case we are given the diameter instead of the radius. We need to find

the radius before we can calculate the area.

6.2

r = = = 3.1mm

Ac π ⋅ r2

(3.14) ⋅ (6.2 ft)2

(3.14) ⋅ (38.44 sq mm)

120.7 sq mm

The area of this circle is approximately 120.7 sq mm.

Example 4

Find the approximate area of the figure.

Solution

We notice that we have two semicircles (half circles) and a small rectangle

The larger radius is 6.2 cm.

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A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle

The smaller radius is

The width of the bottom part of the rectangle is 2.0 cm.

The area of this shape will be

Area of the rectangle:

A = 5.1 ⋅ 2 = 10.2 cm

Area of the outer semicircle π(6.2)2

A = = 9.7 cm

Area of the inner semicircle π(4.2)2

A = = 6.6 cm

A = area of the rectangle +area of the outer semicircle − area of the inner

semicircle = 10.2cm +9.7cm

− 6.6cm = 13.3cm

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1.2.2.1: Exercises

Find the area and circumference of the following figures. a.

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Find the area and perimeter of the following composite figures a.

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1.2.3: Solid Geometric Figures and Objects

6.2.3 Learning Objectives

Find the volume of some common geometric objects

The Meaning and Notation for Volume

The product (length unit)(length unit)(length unit) = (length unit) 3 , or cubic

length unit (cu length unit), can be interpreted physically as the volume of a

three-dimensional object.

Definition: Volume

The volume of an object is the amount of cubic length units contained in the

object.

For example, 4 cu mm means that 4 cubes, 1 mm on each side, would

precisely fill some three-dimensional object. (The cubes may have to be cut

and rearranged so they match the shape of the object.)

Volume Formulas

Figure Volume Formula Statement

The volume of a rectangular solid

Rectangular solid VR l ⋅ w ⋅ h is the length times the width times

(area of base) ⋅ (height)the height.

Sphere 3 The volume of a sphere is

times

s = 3 ⋅π⋅3

times the cube of the radius.

The volume of a cylinder is

Cylinder Cyl 2

times the square of the radius times

(area of base) ⋅ (height)

the height.

Cone c ⋅ π ⋅2⋅ h

Section 1.5.2 Exercises

1. Solve:

b. 2x 3

9 = 10

9x = 13

e. .

Find a unit rate: You bought 10 pounds of potatoes for $4.

Find a unit rate: Joel ran 1500 meters in 4 minutes, 45 seconds.

Which is the more economical purchase: 32 oz of detergent for $6.29

or 48 oz of the same detergent for $8.29?

Solve the proportion.

A student scored 32 out of 40 on a practice exam and 12 out of 15 on

the actual exam. Did the student do better on the practice exam or on the

actual exam?

A soft drink is sold in a six-pack of plastic bottles holding 101.4 fl oz for

$2.69. The same soft drink can be purchased in a 12-pack of cans containing

144 fl oz for $3.79. Which is the better buy?

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