The Relation of Perimeter to Area

Date: 10/2/95 at 14:57:55
From: Kurt S Merz
Subject: ?????

I'm puzzled by the ability of two fences to have the same 
perimeter with very different areas inside them. I realize by LxW 
an 8' x 10' fence will have more area than a 6' x 12' fence, but 
WHY?  Both fences  have 18' surrounding them but different areas.  
Also does a circle or a  square conserve more area with identical 
perimeters?  Thanks for your time.

Shawn

Date: 10/7/95 at 2:33:5
From: Doctor Andrew
Subject: Re: ?????

I confess that your question has stumped me for a good answer.  
Often I've wondered why this is true.  I think it is easier to see 
with a stranger shape than a rectangle.  Imagine that instead of a 
rectangle with straight sides you made the sides all squiggly.  
The rectangle would still have about the same area but the 
perimeter would be much longer than for a simple rectangle.  So we 
can see that perimeter really doesn't need to be related to area 
at all.  In fact, there are shapes called fractals that have only 
a small amount of area and an infinite perimeter (that means that 
no length of string could follow the edge of the shape; pretty 
strange).

>Also does a circle or a square conserve more area with identical 
>perimeters?  Thanks for your Time.

A circle holds more area compared to its perimeter than any other 
shape.

I hope this helps.  Please send us any more questions you have.

-Doctor Andrew,  The Geometry Forum