STATEMENT OF PROBLEM

Our goal in this lesson is to approximate the value of Pi through the estimation of the area of a unit circle. To estimate the area, we will compute the areas of inscribed and circumscribed polygons. As the number of sides of the polygons increases, the area of the polygons will get closer and closer to the area of the unit circle (which is exactly Pi.) This is the method that was used by the Greek mathematician Archimedes over two thousand years ago.

1. Develop a general formula for the area of an n-sided equilateral polygon which is inscribed in a circle of radius r. Hint: Draw a sketch of a typical triangular piece of the n-sided polygon.

2. Develop a general formula for the area of an n-sided equilateral polygon which is circumscribed about a circle of radius r. Hint: Draw a sketch of a typical triangular piece of the n-sided polygon.

3. Using a radius of one, r = 1, generate a table of values which will display the area of the inscribed and circumscribed n-sided polygons. Let the values of n range from 10 to 400. (You need not include every value of n. Instead step by, say, 10 or 20.)

a. How many decimal places of accuracy do you have when n = 400?

b. How many decimal places of accuracy do you have in the average of the inscribed and circumscribed areas?

4.

a. For what value of n will you have one more digit of accuracy than you had in part (3a) in both approximations ?

b. For what value of n will you have six decimal places of accuracy in both approximations?

5. Plot the data in your table on a graph of area versus number of sides. Include in your graph the horizontal line
area=Pi
in order to compare the values of the areas for the inscribed and circumscribed polygons.

6. Which converges faster: circumscribed or inscribed polygons? Justify your answer.