Activities to Find Pi

Date: 10/07/98 at 14:44:08
From: Angie L Peuler
Subject: Math question

I'm a college student taking a Geometry for Elementary Teachers class 
and our professor asked us to find a method for finding Pi other than 
measuring the circumferences and diameters of circles and dividing 
them. I can't seem to find anything in the books that I have. Can you 
suggest any other methods?

Thank you,
Angie Peuler

Date: 10/19/98 at 21:57:26
From: Doctor Santu
Subject: Re: Math question

Hello Ms. Peuler:

Pi is everywhere! You can hardly do any advanced mathematics for
fear of encountering Pi. However, practical methods for measuring Pi
usually do come from geometry or physics.

There are ways of calculating Pi *in principle*. In other words, they 
are difficult to use in practice, because they always involve an 
infinite number of steps. One method is to calculate as many terms of 
the infinite sum:

4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 - 4/15 + 4/17 - 4/19 + 4/21...

as you can. It comes as close to Pi as you want. The error is 
approximately the size of the first term you leave out. For instance if 
you add these numbers up until you get to 4/40000000001, your answer 
would be off by less than 1 in 10000000000, which is excellent! (But 
who has the time, eh?)

Does weighing a steel ball count? This involves physics, but your 
instructor might like it. You have to use two weighings. Here's the 
plan.

Get a perfect steel ball with a little hook attached, and carefully 
weigh the ball, for instance by hanging it from one arm of a chemical 
balance (you know those things that look like the scales of justice).  
Weigh it using grams, not ounces, or the calculations become hairier.

Now measure the diameter of the ball carefully. (Yes, unfortunately, 
you have to measure the diameter of the ball! Physics people have 
calipers that you can use for this, called vernier calipers.) Suppose 
that you figure the radius (half the diameter) is R centimeters.

Now, you have to weigh the ball again, but this time, it must be 
hanging inside a beaker of water. You must hang the ball from the arm 
of the balance, again, inside a container of water. It mustn't touch 
the bottom or the sides of the container.

      .______T_______ 
      |             /\
   |--|--|         /  \
   |  O  |        / xx \
   |_____|        ------

Hope that gives you an idea about the setup.

Now this time the ball should actually weigh less. The reduction 
should be exactly the weight of a ball of water the same size as the 
steel ball! So, suppose the reduction is W grams. It so happens that W 
grams of water has a volume of exactly W cubic centimeters. (For water, 
1 gram = 1 cubic centimeter.)

A well-known formula says that the volume of a ball of radius R should 
be (4/3)*Pi*R^3, which you have measured to be W.

Multiply by 3, and you get 4*Pi*R^3 = 3W.

Since you know R, Pi should be 3W divided by (4*R^3).

The general idea here is not very different from measuring the 
circumference of a circle and dividing by a diameter, so it might not 
qualify as a truly different idea. By the way, it is much more 
convenient to measure the diameter of a cylinder, such as a tin can.

You must realize that all of these only give you approximations for 
Pi. In cases where we are measuring anything, there is always error.  
In the case of the infinite series that you add up, there is the error 
due to not being able to add the whole series.

If you would be satisfied with a really excellent approximation, you 
could use 355/113, which is a lot better than 22/7.

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