Pi and Polygons

Date: 03/14/99
From: Anonymous
Subject: Pi and Polygons

A simple iterative derivation inscribes an n-polygon inside a circle 
until the limit approaches pi (the circumference of the unit circle).

Is there a formula to find the angle of an n-sided polygon given 
that it has x sides? This formula is obviously convergent. Take the 
limit as theta -> 180 (or pi radians) and the infinity*0 = pi.?

Date: 03/14/99
From: Doctor Ken
Subject: Re: Pi and Polygons

This method of finding successive approximations to Pi is one of the 
oldest methods known, because it is one of the easiest to understand, 
and you can draw nice pictures to explain it.

If a regular polygon has n sides, then we can draw lines from its 
center to all the vertices, and these lines will divide the pie-shaped 
picture into n wedges. Each wedge's central angle will have a measure 
of 360/n degrees. So, we can draw this picture of one wedge, where C 
is at the center of the polygon:

               A
             /|
           /  |
         /    |
       /      |
     /        |
 C /__________| D
   \          |
     \        |
       \      |
         \    |
           \  |
             \|
               B

Segment AB here is one side of the original regular polygon. 
Since angle ACB is 360/n, angle ACD is 180/n. Therefore, if the length 
of AC is 1/2, the length of AD is sin(180/n)/2. Therefore, the length 
of AB is sin(180/n), and the perimeter of the entire polygon is 
n*sin(180/n).

So, you are correct: when you let n go toward infinity, sin(180/n) 
will tend towards zero. Since we know that a circle whose radius is 
1/2 has a circumference of Pi, the 0 and infinity balance each other 
in the limit.

Of course, there is a practical problem with all of this. In order to 
calculate the sines, you need to know a thing or two about Pi. It is 
true that for some special angles like 30, 45, and 60 degrees (and 
their sums and differences, etc.) you can write down an explicit 
elementary expression for their sines, this is not true of some other 
angles like 180/11.

So, how do we calculate the perimeters without knowing Pi already? We 
need to find some trick, or we need to find some other method entirely 
of approximating Pi. And that is where a great part of the glorious 
history of mathematics starts.

For more about Pi, its role in history, and the various attempts to 
know it better, check out the excellent book _A History of Pi_ by Petr 
Beckmann. People have done some pretty clever things to get to know 
Pi.

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