Finding Pi: Buffon's Needle Method

Date: 1/31/96 at 19:45:10
From: Robert Garry
Subject: Calculating PI

Dr. Math,

For my Survey of Math class, the students are calculating the 
value of PI.  They have come up with several methods to do so, and 
I was hoping to show them a surprising way using a needle and 
parallel lines.  You drop the needle and record if it touches a 
line... Could you give me a reference for this method?

Mahalo,

Robert Garry

Date: 7/13/96 at 14:48:8
From: Doctor Jodi
Subject: Re: Calculating PI

Hi Robert!  It took a while to find the name of this method.  It's 
called Buffon's needle problem.  The net has many descriptions of 
it, including this one from 

http://www.mste.uiuc.edu/reese/buffon/buffon.html     

Here's the text (but please visit that site for the pictures, if 
you can).

Introduction

Buffon's Needle is one of the oldest problems in the field of 
geometrical probability. It was first stated in 1777. It involves 
dropping a needle on a lined sheet of paper and determining the
probability of the needle crossing one of the lines on the page.  
The remarkable result is that the probability is directly related 
to the value of pi. 

These pages will present an analytical solution to the problem 
along with a program (written for Macintosh computers) for 
simulating the needle drop in the simplest case scenario in which 
the length of the needle is the same as the distance between the 
lines. 

The Simplest Case

Let's take the simple case first.  In this case, the length of the 
needle is one unit and the distance between the lines is also one 
unit. There are two variables, the angle at which the needle falls 
(theta) and the distance from the center of the needle to the 
closest line (D). Theta can vary from 0 to 180 degrees and is 
measured against a line parallel to the lines on the paper. The 
distance from the center to the closest line can never be more 
that half the distance between the lines. The graph below depicts 
this situation. 

The needle in the picture misses the line.  The needle will hit 
the line if the closest distance to a line (D) is less than or 
equal to 1/2 times the sine of theta. That is, D <= 
(1/2)sin(theta). How often will this occur? 

In the graph below, we plot D along the ordinate and 
(1/2)sine(theta) along the abscissa. The values on or below the 
curve represent a hit (D <= (1/2)sin(theta)). Thus, the 
probability of a success it the ratio shaded area to the entire 
rectangle. What is this to value? 

The shaded portion is found with using the definite integral of 
(1/2)sin(theta) evaluated from zero to pi. The result is that the 
shaded portion has a value of 1. The value of the entire rectangle 
is (1/2)(pi) or pi/2. So, the probability of a hit is 1/(pi/2) or 
2/pi. That's approximately .6366197. 

To calculate pi from the needle drops, simply take the number of 
drops and multiply it by two, then divide by the number of hits, 
or 2(total drops)/(number of hits) = pi (approximately). 

The Other Cases

There are two other possibilities for the relationship between the 
length of the needles and the distance between the lines. A good 
discussion of these can be found in Schroeder, 1974.  The 
situation in which the distance between the lines is greater than 
the length of the needle is an extension of the above explanation 
and the probability of a hit is 2(L)/(K)pi where L is the length 
of the needle and K is the distance between the lines. The 
situation in which the needle is longer than the distance between 
the lines leads to a more complicated result. 

Questions

1. After 1,000 drops, how close would you expect to be to pi?

2. After 264 drops, the estimate of pi is 3.142857. This estimate 
   is correct to within 2/1000 of the book value of pi. Will the 
   next drop:
      A. make the estimate more accurate?
      B. make the estimate less accurate?
      C. make it more or less accurate depending on whether it's 
         a hit or miss? or
      D. impossible to say. 

3. What about the next 10 drops?

References

Cheney, W. and Kincaid, D. (1985). Numerical Mathematics and 
   Computing. 2nd Ed. Pace Grove, California: Brooks/Cole 
   Publishing Company pp. 354-354 

Schroeder, L. (1974). Buffon's needle problem: An exciting 
   application of many mathematical concepts. Mathematics Teacher, 
   67 (2), 183-186. 

Anoother resource on this topic: a lesson plan on calculating pi at 

http://www.mste.uiuc.edu/undergradmathed/mathletes/daleleibforth/pi2.html   

Thanks for your question and let us know if we can help again!

-Doctor Jodi,  The Math Forum