Two Unusual Ways of Estimating Pi

Date: 03/24/97 at 21:55:10
From: Phydeaux
Subject: Two Unusual Ways of Estimating Pi

Hello, Dr. Math,

I have to write a detailed five page report on the history of Pi 
and two ways to calculate it. The history of pi is no big deal to me.  
It's the ways to estimate pi that is bugging me.  

I have found the information on Buffon's needle problem but I'm 
looking for something VERY unusual. I've also discovered (through 
Buffon's needle problem) that pi is associated with things that don't 
have anything to do with circles or spheres.  Help me out, please.  
As I said, I'm looking for two unusual ways to estimate pi.

Muchas gracias,

   Tim Dwyer

Date: 03/25/97 at 14:49:12
From: Doctor Steven
Subject: Re: Two Unusual Ways of Estimating Pi

Pi is defined as the ratio of the area of a circle to its radius 
squared. If you inscribe polygons with the same radius as the circle, 
then their area is smaller than the circle.  Find the ratio of their 
area to the radius and you have an estimate that is too small for Pi.  
If you circumscribe polygons with the same radius as the circle then 
they have an area that is larger than the circle.  Find the ratio of 
their area to the radius squared and you get an estimate that is too 
high for Pi.  You've now squeezed Pi between two values.  The more 
sides you add to your polygons the better the approximations, and the 
tighter the interval where you know Pi to be. (Hint: if you let the 
radius equal one then the area of the polygons should approximate Pi).

The probability that an integer picked at random will have repeated 
prime factors is 6/PI^2.  Using the 100 million decimal place 
expansion of Pi, take every 100 digits as an integer, find whether it 
has repeated prime factors.  The proportion of integers that do should 
equal 6/Pi^2.  Using this method "researchers" have found the value of 
Pi to about 20 decimal places.  (Not a very useful approximation 
method for Pi, but a very neat one.)

Hope this helps.

-Doctor Steven,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/