Number of Points in a Star

Date: 7/16/96 at 23:36:33
From: Anonymous
Subject: Number of Points of Star

Dear Doctor Math,

Can you solve this problem?  We (with Rachel - 9 and David - 6) are 
sending this question to you because it came up in my daughter's 
fourth grade class and neither the teacher or any parents could figure 
it out.  My father is a civil engineer and he used to teach calculus, 
but he just said that "it is not a trivial problem."

They were using LOGO to make stars and other shapes.  They made stars 
with commands like this:

  repeat 24[fd 200 rt 165]

The teacher noticed that the points of the star weren't directly 
related to the internal angle of the points, but there were some 
interesting corollaries. She wondered if there was a way to predict 
the number of points in a star given only the internal angle of the 
corners (assuming that to be a constant).

Any ideas on the answer, and additionally, any ideas on working that 
into a lesson of some sort (especially in the context of a LOGO 
section)?

Thank you,

Stephen (and Rachel and David) Saxon.

Date: 7/17/96 at 18:49:40
From: Doctor Anthony
Subject: Re: Number of Points of Star

The sum of the interior angles at the points of the star should total 
180 degrees whatever the number of points that the star has. 

You can persuade yourself of this by going to a point, facing along a 
line towards the star and rotating anticlockwise through the interior 
angle at the point. Now proceed to the next point along a straight 
line (I am assuming this is how your star is created) and turn about 
and rotate anticlockwise through the interior angle at the point.  
Continue on to the next point and repeat the procedure. When you 
finally return to the starting point, however many points there are, 
you will have rotated through a total of 180 degrees when rotating 
anticlockwise through the interior angle at each point.

Sum of the 'point' angles = 180 degrees.

So if there are 5 points each is 36 degrees.  If six points, each is 
30 degrees and so on. 

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