Sum of the Angles in an N-Pointed Star

Date: 11/29/1999 at 19:58:50
From: Ashley
Subject: Equations for star polygons

I am doing a math project for Honors Geometry. We were given two stars 
and told to find the sums of the angles at the tips of each of the 
stars. We found the 5-pointed star to be 180 degrees and the 6-pointed 
star to be 360 degrees. 

Then we were told to find a formula for the angle measures at the tip 
of an n-pointed star. That is the part we are stuck on. The way we 
figured out the 5-pointed star was from a computer program. We figured 
out the 6-pointed star because it is 2 triangles, and the sums of 
their angles are 180 and 180, which is 360 degrees. So, I guess we 
just need to find the formula for the sum of the angle measures.

Thank you.

Date: 11/30/1999 at 13:55:28
From: Doctor Rob
Subject: Re: Equations for star polygons

Thanks for writing to Ask Dr. Math, Ashley.

There are two kinds of 7-pointed star. If you number the points 
cyclically around the center as 0, 1, 2, 3, 4, 5, and 6, then you get 
one kind by connecting them in the order 02461350, and the other by 
connecting them in the order 03625140. The points on the latter are 
sharper, and the angles add up to 180 degrees. The points on the 
former are less sharp, and add up to 3*180 = 540 degrees. 

For an 8-pointed star, there are also two kinds, whose point angles 
add up to either 2*180 or 4*180 degrees. 

For a 9-pointed star, there are three kinds, whose point angles add up 
to 1*180, 3*180, or 5*180 degrees.

Are you beginning to see a pattern?

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/