Geometry

Proof Without Words: The Vertex Angle Sum of a Regular Star Polygon

Matthew Jakubowski ([email protected]), University of Delaware, Newark, DE, and Raymond Viglione ([email protected]), Kean University, Union, NJ

The regular star polygon { p/q} is constructed by connecting every qth of p evenly spaced points on a circle, where p and q are positive integers satisfying 1 < q < p/2.

Theorem. The sum of the measures of the vertex angles of { p/q} is 180 p − 360q. Proof. We demonstrate the general proof technique on {7/2}.

180 p

p 360

180 pq •

180 pq•

180 –p • 180p2q •

Compare [2] for the case of five points and [1] for when the points are not evenly spaced on the circle.

Summary. We visually prove the formula for the vertex angle sum of a regular star polygon.

References

1. C. Alsina, R. Nelsen, Icons of Mathematics. Mathematical Association of America, Washington, DC, 2011.

2. F. Nakhli, Behold! The vertex angles of a star sum to 180◦, College Math. J. 17 (1986) 338, http:// dx.doi.org/10.2307/2686283.

http://dx.doi.org/10.4169/college.math.j.46.2.109 MSC: 97G20

VOL. 46, NO. 2, MARCH 2015 THE COLLEGE MATHEMATICS JOURNAL 109