Archimedes on Pi

One of the most remarkable coincidences in the history of mathematics is that , the ratio of the circumference of a circle to its diameter, is also equal to the ratio of the area of a circle to the area of the square whose sides are equal to the radius. We write these relationships as

These formulas are so well known that few pause to wonder why this same strange irrational we denote by should appear in both formulas.

Almost certainly, was first known as the ratio of the circumference to the diameter. Even this fact must have been surprising: No matter what circle, no matter what size, this ratio is always the same. If you want to construct a circle of a certain circumference, you can use this constant ratio to determine what the diameter should be. If you know the diameter, you can use it to find the circumference.

Area is a more complicated concept than length, but the time came when people asked how to construct a circle of a given area. What should the diameter be? If you know the diameter, what will be the resulting area? Imagine their surprise when they compared the area of a circle to the area of the square whose sides are the length of the radius and discovered that it looked like it was exactly the ratio of the circumference to the diameter

.

In some respect, this shouldn't be surprising. If we consider a regular polygon, it is made of of isosceles triangles such as triangle OAC in the figure shown above. since OB is perpendicular to AC, the area of this triangle is (1/2) OB AC. If we have n sides to our polygon, then it is made up of n congruent triangles, and the total area is (1/2) OB AC n. But AC n is the circumference of the polygon, so its area is (1/2) inner radius • circumference. As the number of sides, n, gets larger, the polygon approaches a circle and the inner outer radii converge to a single value of the radius.If r is the radius of the circle, then its area should also be

Archimedes, in On the Circumference of the Circle, uses this comparison with inscribed and circumscribed polygons to show that the area of the circle cannot be large than or smaller than He then uses this comparison with the areas of polygons to show how to approximate the vlue of .