Angles of Stars

Date: 08/18/97 at 01:07:22
From: Special Ed
Subject: Practical geometry

What are the interior and external angles of stars built on the 
following regular polygons: a pentagon and octagon. I don't need 
lengths, just the angles. Or how about just the interior angles of a 
regular pentagon and an octagon? I have found definitions, but not the 
angles.

Thank you for helping a frustrated parent. Where else should I have 
looked?

Date: 08/22/97 at 14:43:06
From: Doctor Rob
Subject: Re: Practical geometry

To answer the last part first, the interior angles of a regular 
pentagon have measure 108 degrees, and those of a regular octagon have 
135 degrees.

To build a star on a regular n-sided polygon, you would add n 
congruent isosceles triangles whose bases are the same length as the 
sides of the polygon, by joining each side of the polygon to the base 
of one of the isosceles triangles. There is some freedom here, in that 
you can make the other sides of all the isosceles triangles have 
length equal to anything you like, as long as it is more than half the 
side length of the polygon. For example, you could make all the 
triangles equilateral.

I think this option looks poor for a pentagon. The usual five-pointed
star is formed by extending the sides of the pentagon until they meet
outside the pentagon.  In this case, the base angles of the triangles
have measure 72 degrees, and the vertex angle has measure 36 degrees.
When you erase the original pentagon, the large interior angles are
252 degrees, and the exterior angles are 108 degrees.

In the case of an octagon, one can extend the diagonals of the octagon
until they meet, in which case the vertex angle of the points has 
measure 45 degrees, and the base angles 67.5 degrees each. When you 
erase the original octagon, the large interior angles are 270 degrees, 
and the exterior angles are 90 degrees.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/