Equable Polygons and the Area of a Circle

Date: 10/30/2001 at 14:00:55
From: suzi-mary
Subject: Area of circle and pi

I have been researching equable regular polygons and have found a 
general formula to work out the length of side the polygon (of any 
shape) must have to be equable. Apparently this is closely linked to 
finding an equation for the area of a circle without using pi (and 
therefore proving pi). If you could help me in any way, I would be 
eternally greatful.

Thanks.

Date: 11/01/2001 at 11:25:09
From: Doctor Roy
Subject: Re: Area of circle and pi

Hello,

Thanks for writing to Dr. Math.

The method is quite simple. Find the distance from the center of a 
regular polygon to any of the vertices. Call this distance r. Then 
calculate the area of the polygon. Find a relation between the area 
and r^2, and you will get an approximation for pi.

I hope this helps.

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

Date: 11/01/2001 at 12:49:32
From: Doctor Peterson
Subject: Re: Area of circle and pi

Hi, Suzi-Mary.

I think what you have in mind may be the idea hinted at here:

   Finding a Circle's Area Without Pi
   http://mathforum.org/dr.math/problems/anna.2.24.01.html   

The formula for an area that doesn't use pi is

    A = Cr/2

where C is the circumference and r is the radius. This is also true 
for polygons, if you use the right "radius"; see this answer for more 
details on doing this with polygons, and also on what it proves about 
pi:

   Why Pi?
   http://mathforum.org/dr.math/problems/crystal.01.25.01.html   

Using this formula makes it easy to find the size of an "equable" 
polygon or circle! I hadn't thought of this approach myself.

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