Handout 15B

The purpose of this exercise is to study different methods of approximating the number Pi. The topic was suggested by Bonnie Saunders; the information provided by Doctor Wilkinson and the Math Forum ( http://www.forum.swarthmore.edu/dr.math/problems/paradis5.20.98.html) was also very helpful.

As we all know, the area of a circle with radius r is given by the formula Pi r^2. We also know that the value of Pi is something like 3.14. But how do we know that?

Today, the value of Pi is well known. Two thousand years ago, this wasn't the case. Around the year 250 BC, the Greek mathematician Archimedes had to figure out the value of Pi for himself. Here's how he did it:

Hexagons inscribed in and circumscribed about a circle.

He knew that Pi was area of a circle with radius one. He knew how to find the area of a regular polygon of a certain size, but not the area of a circle. So, he found the area of a regular polygon with 96 sides; this is very close to the area of a circle. However, he wasn't content to stop there. Because he wanted some idea of how far wrong he might be, he put one 96-gon inside his circle and computed its area, and one 96-gon around his circle, and computed its area. (See the figure above.) The area of the circle had to be somewhere between the areas of the two 96-gons.

Using a little trigonometry, we find that when we inscribe a regular polygon with n sides in a circle of radius one its area is:

n/2 sin(360/n)

The area of a regular polygon with n sides that is circumscribed about a circle of radius one is:

n tan(180/n)

The diagram above illustrates that 2.60 < Pi < 3.46. When n is 96, the fact that the area of the inscribed polygon is less than Pi which is less than the area of the circumscribed polygon gives us Archimedes' estimate of Pi:

3.1394 < Pi < 3.1427

But wait! It would have been as hard for Archimedes to calculate sin(360/96) as it was to compute Pi! How did he find these areas?

We're not going to follow the steps Archimedes actually used to compute Pi, but we can use LOGO to write an attractive demonstration of his results.

Write a program that takes two arguments, :RADIUS and :N and draws the inscribed and circumscribed regular :N-gons of a circle of radius :RADIUS. Have the program print the lower and upper bounds on the value of Pi given by the equations above.

If you have extra time, you can draw the circle between the two :N-gons; this will be easier for you if you have written one or more separate procedures to draw :N-gons. Or you can use the WAIT, CS and FOR commands to make a LOGO "slideshow" giving closer and closer approximations to Pi.

After the development of calculus in the late seventeenth century, more algebraic methods of computing Pi became available. Machin's forumla states that Pi/4 = 4 arctan(1/5) - arctan(1/239), and using Taylor series from calculus one can compute that:

arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...

Combine these two formulas to write a LOGO program that estimates Pi by computing the first several terms of this Taylor series for arctan(1/5) and arctan(1/239). Of course, you will want your program to have an argument indicating how many terms to calculate. You will probably also want to write a separate procedure to compute the approximation of arctan(x).

Compare this approximation to the one used by Archimedes. Which do you think is better? Why?

There are many more methods for estimating Pi, and computers have been used for this project practically since they were invented. The value of Pi is now known to billions of decimal places. A finite exact decimal value for Pi does not exist, but we can get approximations as close to exact as we could conceivably need.

Mtht420