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### 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.

```

```
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/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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