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h__5._ch._8.pdf
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GEOMETRY Ch. # 8
8.1 Perimeter and Area
Objectives
1. Find the perimeter of basic polygons, including squares, rectangles, and triangles 2. Apply the Pythagorean Theorem to a right triangle 3. Find the area of basic polygons, including squares, rectangles, triangles,
parallelograms, and trapezoids
4. Apply Heron’s Formula to find the area of a triangle 5. Use in the calculation of circumference and area of circle 6. Understand and apply the units that commonly used in astronomical measurement,
including astronomical units, light-years, and parsecs.
Polygon
A polygon is a closed plane figure whose sides are line segments that intersect only at
the end points.
Perimeter
The perimeter of a two-dimensional figure is the sum of the lengths of its sides.
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1. A quadrilateral is a polygon that has four sides. 2. A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. 3. A kite is a quadrilateral with two distinct pairs of congruent adjacent sides. 4. A rectangle is a parallelogram that has a right angle. 5. A square is a rectangle that has two congruent adjacent sides 6. A rhombus is a parallelogram with two congruent adjacent sides. 7. A trapezoid is a quadrilateral with exactly two parallel sides.
Area
The area f a two-dimensional figure is the number of square units it takes to fill the
interior of that figure.
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1. The area A of a rectangle whose base length b and whose altitude has length h is given by: A b h
2. The area A of a square whose sides are each of length S is given by: 2A S 3. The area A of a parallelogram with a base of length and with corresponding altitude
of length is given by: A b h
4. The area of a triangle whose base has length and whose corresponding altitude has
length is given by: 1
2 A b h
5. Heron’s Formula: If the three sides of a triangles , ,a b and c then the area A of the
triangle is given by: ( )( )( )A s s a s b s c , where 12s a b c 6. The area of a trapezoid whose base has lengths and whose altitude has length is
given by: 1 2 1
2 A h b b
7. The area A of any quadrilateral with perpendicular diagonals of lengths 1 2
d and d is
given by: 1 2
1
2 A d d
8. The area A of a rhombus whose diagonals of lengths 1 2
d and d is given by:
1 2
1
2 A d d
9. The area A of a kite whose diagonals of lengths 1 2
d and d is given by: 1 2
1
2 A d d
10. The ratio of the circumference of a circle to the length of its diameter is a unique positive constant.
11. is the ratio between the circumference C and the diameter length d of any circle;
thus C d
in any circle.
12. The circumference of a circle is given by the formula: C d
or 2C r
13. The area A of a circle whose radius is r given by: 2A r
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A circle is the set of all points in a plane that are at a fixed distance from a given point, the center of the circle.
A radius is a segment that joins the center of the circle to a point on the circle.
All radii of a circle are congruent.
A line segment that joins two points of a circle is a chord.
A diameter of a circle is a chord that contains the center of the circle.
In a circle, the length of a diameter is twice that of a radius.
Arc is the segment (part) of a circle determined by two points on the circle and all points between.
Pythagorean Theorem
In a right triangle with hypotenuse of length c and legs lengths a and b ,
2 2 2
c a b
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8.2 Volume and Surface Area
Objectives
Find the Volume of basic three-dimensional objects, including rectangular solid, cylinders, spheres, cones and pyramids.
Find the surface area of basic three-dimensional objects, including rectangular solids, cylinders, and spheres.
Volume
Volume is the amount of space occupied by
a three-dimensional object.
The volume of a right rectangular prism is
given by: . .V l w h where l measures the length, w the width, and h the altitude
(height).
Volume of a Figure having identical Cross Sections
If a three-dimensional figure has height h and identical
cross sections from top to bottom, each of area A ,
the volume V Is: .V A h
Surface Area
The surface area of three-dimensional figure is the sum
total of the areas all the surfaces that compose the figure.
Sphere
A sphere is the three-dimensional counterpart of a circle.
It is the set of all points in space equidistant from a fixed
point.
That fixed point is called the center of the sphere, a line
that connecting the center and any point on the sphere is a
radius and a line connecting any two points of the sphere and passing through the
center is a diameter.
The length of the diameter is twice that pf the radius.
Volume and Surface Area of a Sphere
The Volume V of a Sphere of radius r is: 34
3 V r
The surface area A of a sphere whose radius r is: 2
4A r
Volume of a Cone or Pyramid
The volume V of a pyramid or a cone having a base area A and an altitude of length h
is given by: 1
3 V A h
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Ex:
Find the volume and the surface area round to nearest two decimal places.
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8.9 Fractal Geometry
Objectives
Understand how a recursive process can be used to generate fractal
See how self-similarity occurs in both the world and fractal geometry
Be able to comprehend and compute a fractal’s dimension
Grasp that some real-world objects have fractal dimensions
A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern
that displays at every scale. If the replication is exactly the same at every scale, it is called
a self-similar pattern.
The sierpinski gasket
The sierpinski gasket is one of the older examples of fractal geometry.
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The sierpinski gasket is the result of continuing the process without stopping.
The enlarged portion would look exactly the same as the non-enlarged portion; this is called
self-similarity.
A shape has self-similarity if parts of that shape appear within itself at different scales.
Fractal geometry utilizes shapes that have self-similarity.
Some shapes have exact self-similarity (In which the enlarged portion is exactly the same as
the non-enlarged portion), and some shapes have approximate self similarity.
Recursive Processes
Recursion is the process of repeating items in a self-similar way.
