Proving Pi and Buffon's Needle

Date: 10/19/97 at 14:40:47
From: Brandon Billings
Subject: Pi

I have to write a paper on pi and need books on pi. I went to the 
library and got some books but they didn't give me the information 
that I needed. What experiment can I do to prove pi using both 
mathmatics and science??

Date: 11/08/97 at 16:30:43
From: Doctor Sonya
Subject: Re: Pi

Hi Brandon,

Pi is a great thing to have to write a paper on. Here are two ways to 
determine the value of pi using scientific experiments. One, from Dr. 
Sonya, is very easy, and the other, from Dr. Mark, is very complicated 
(but way cooler!).

Dr. Sonya says:

You can estimate pi using a very common formula. You probably know it 
already. It says that if you have a circle with radius r, its 
circumference is 2(pi)r.  If you look at the ratio of the 
circumference to the radius, C/r, you get:  

   C   2*pi*r
   - = ------ = 2pi.  
   r      r

This means that if you take a circle, measure its radius and 
circumference, and find C/r, you will get a number equal to 2*pi.  
Then you just have to divide this number by 2 to get an estimated 
value for pi.  

(A hint for doing this experiment: to measure the circumference of a 
circle, lay a piece of string all the way along the edge, trimming it 
so that there are no extra ends hanging off. Then straighten it out 
and measure how long it is. Do you see why this is the same length as 
the circumference?) 

Dr. Mark says:

I hope that one of the books you got from the library was Petr 
Beckmann's <i>A History of Pi,</i> which is the authoritative book on 
the subject. If you look on page 159 of that book, you will find a 
description of an experiment you could do to determine pi. The 
experiment was thought up by George Louis Leclerc, the Comte (Count) 
de Buffon, and is known as "Buffon's needle."  Here's how it goes:

Put parallel lines a distance d apart on the floor (or any horizontal
surface: you could get a big piece of paper, draw lines on that, and 
tape it to the floor, that way no one will yell at you for drawing 
lines on the floor), and drop a needle (or a straw, or a toothpick) of 
length L onto these lines. You need to have d bigger than L for this 
to work.  

If you do this N times, and you find that the needle (or toothpick, 
or...) touches one of the lines (it can't touch both: why?) K times, 
then pi = (2LN)/(Kd). Of course, you only get an approximation to pi, 
but the approximation is reasonable if N is big (a couple of hundred 
or so). The explanation for why this works is not easy to give, since 
it involves calculus, but you can find that explanation in Beckmann's 
book.

You will have to play around with the experimental arrangement (how 
many lines to use, how high up the needle should be when it is 
dropped, and so on) to see which gives the best results, but usually, 
choosing d just a little bigger than L works best.

If you do find Beckmann's book, look also at page 164, where you will 
find a short computer program (written in the BASIC computer language) 
which simulates the throwing of the needle onto the parallel lines.  
In one run of this program, it took about 3500 throws to get a value 
for pi of 3.14, so unless you are planning to spend a lot of time 
doing this experiment, you shouldn't expect to get terribly close to 
the correct value of pi.

-Doctors Mark and Sonya,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/