Thinking about the Maximum Area Enclosed by a Fence

Date: 04/15/2004 at 06:11:44
From: Jamie
Subject: The Fencing Problem! You have 2000 meters of fencing...

You have 2000 meters of fencing.  What is the largest area you can 
enclose with it using various shapes?  For example, you could make a
square with sides of 500 meters, which would give an area of 500*500 =
250,000 sq meters.  But is that the biggest possible area?

I just don't know where to start.  I mean you could just guess shapes 
and work them out but there must be some kind of pattern you need to 
follow to get it done!

Date: 04/15/2004 at 12:43:21
From: Doctor Peterson
Subject: Re: The Fencing Problem! You have 2000 meters of fencing...

Hi, Jamie.

To actually prove that a particular answer is correct takes some 
pretty advanced math; but you should be able to find the answer and 
convince yourself it's right by considering a few "patterns":

1. What happens if you compare a regular polygon with an irregular one 
of the same type?  For example, compare the areas of several 
rectangles with that of a square with the same perimeter; or compare a 
rectangle with a parallelogram with the same side lengths.  That will 
tell you what kinds of angles and sides work best.

2. What happens to the area if you increase the _number_ of sides of a 
polygon while keeping the perimeter the same?

3. And, if your fencing is flexible rather than in straight sections, 
what happens if you try some simple curved shapes?

In other words, although you _will_ be trying various shapes, you will 
be not _just_ be randomly trying to find the biggest, but rather 
looking for patterns in the results you get that will lead you in the 
right direction.

This is like playing "20 questions" or a similar guessing game, and 
not just randomly asking specific questions like "Is it the Great
Pyramid?" or "Is it a Volkswagen Beetle?", as a young child will, but
rather starting with general questions like "Is it man-made?" or 
"Would it fit in a garage?", and working systematically in the 
direction of the right answer.

If you need area formulas for the shapes you want to try, see our FAQ 
under "Formulas".

If you have any further questions, feel free to write back.

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