Regular and Non-regular Polygon Areas

Date: 03/10/99 at 17:10:37
From: Robert Davies
Subject: Regular Polygons

I am trying to work out the proof that regular polygons give the 
maximum area but as of yet have not succeeded.

Please help!

Date: 03/11/99 at 11:55:35
From: Doctor Peterson
Subject: Re: Regular Polygons

The first approach that comes to mind is to take any non-regular 
polygon and show that you can find a larger polygon with the same 
perimeter. Then the largest polygon with that perimeter must be 
regular.

Suppose that there are three consecutive vertices A, B, and C in a 
polygon such that AB and BC have different lengths. See if you can find 
a point B' where AB' and B'C are the same length, but their sum is the 
same as AB + BC. Then show that the area of triangle AB'C will be 
larger than that of ABC. If you replace B with B' in the polygon, its 
perimeter stays the same but the area is larger.

                       B'
             B         +
             +       /   \
            /   \  /       \
           /     /  \        \
          /    /        \      \
         /   /             \     \
        /  /                   \   \
       / /                        \  \
    A //                              \\  C
     + - - - - - - - - - - - - - - - - - +
     |                                    \
     |                                     \
     ...                                    ...

This will show that the largest polygon has to have all sides the same, 
since a polygon whose sides are not the same is never the largest.

You will also have to show that the angles in the largest polygon with 
a given perimeter are all the same. Try a method similar to what we 
just did for the sides. (I have not taken the time to work that part 
out.)

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