Interior Angles of a Polygon

Date: 05/20/97 at 19:46:32
From: Danielle Deitz
Subject: Interior angles of a polygon

What can you conclude about the sum of the interior angles of a 
pentagon? A triangle? A quadrilateral? A hexagon?

Does the sum depend on whether or not the polygon is convex? Why?

What would you predict the sum of the interior angles of a 20-sided 
polygon to be? How do you calculate this? Would the calculation be 
20 x 360 because there are 360 degrees in a quadrilateral? 

Can you please help me?

Date: 05/21/97 at 10:59:15
From: Doctor Wilkinson
Subject: Re: Interior angles of a polygon

You had a good idea there to work from a simple case up to the big 
polygon. Unfortunately, your guess was off because you didn't see just 
how the smaller case relates to the big one.

Let's look at the simplest cases first. For a triangle, there's an
important theorem that tells you that the sum of the interior angles 
is 180 degrees. From this you can figure out the rest.

Let's look at a rectangle next. Here you can see the answer in two 
ways. First, you know that all the angles of a rectangle are 90 
degrees, and there are four of them, so the sum is 4 * 90 = 360 
degrees.  

Let's look at this another way.  Draw a diagaonal across the rectangle 
to divide it into two triangles:

      --------
      |\     |
      | \    |
      |  \   |
      |   \  |
      |    \ |
      |     \|
      --------

The sum of the interior angles of each triangle is 180 degrees, and 
all the interior angles of the triagle make up the interior angles of 
the rectangle. Since there are two triangles in the rectangle, the 
sum of the interior angles of the rectangle is 2 * 180 = 360 degrees.

This argument doesn't depend on having a rcctangle. Any convex
quadrilateral will do. If you draw the right diagonal, you can make it
work for even a non-convex quadrilateral.

Now see what you can do for a pentagon. Again you can start at one 
vertex and draw two diagonals. This divides the pentagon into three 
triangles, and you can see that the sum of the angles of the pentagon 
is 3 * 180 = 540 degrees.

Now we should begin to see a pattern.  Let's make a table:

     number of sides  sum of interior angles
     ---------------  ----------------------

            3                1*180
            4                2*180
            5                3*180
           ....
           20                  ?

The part about whether it makes a difference if the polygon is convex 
or not is the trickiest question. The answer is no, but this is not 
completely obvious. The idea is that no matter how the polygon twists 
and turns, you can always cut off a triangle that stays entirely 
inside the polygon, so that what is left is a polygon with one fewer 
side.

The general idea here, as in a lot of math problems, is to start with 
the simplest cases, where you can see what is happening, and work your 
way up to more complicated cases, looking for a general pattern.

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