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Sum of Star Angles


Date: 12/19/2001 at 07:04:13
From: Jake Heeter
Subject: Geometry (angles of a polygon)

There is a picture of a distorted star and I was asked to find the 
sum of the measure of the angles formed at the tips of each star.  I 
would be able to do this but they give no measurements. I don't even 
know where to start.


Date: 12/19/2001 at 08:30:15
From: Doctor Peterson
Subject: Re: Geometry (angles of a polygon)

Hi, Jake.

There is a formula for the sum of the interior angles of an n-gon that 
does not require that it be regular, or that anything be known about 
it other than the number of sides. This can be extended to what we can 
call an n-gram, an n-pointed star, by taking account of the number of 
times the star winds around the center.

Think first about the EXTERIOR angles. These are the angles through 
which you have to turn at each point if you "walk around" the star. 
Imagine starting at one point, and walking all the way around until 
you get back to the same point and are facing the same direction you 
started. Then you will have rotated through an angle equal to the sum 
of the n exterior angles. In the case of a five-pointed star, you will 
have turned around not once (as you would do in going around a 
pentagon), but TWICE. Do you see why? So the sum of the exterior 
angles is 2*360 = 720 degrees.

Can you use this to determine the sum of the interior angles?

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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