Archimedes' Method of Exhaustion

In Measurement of the Circle, the great Archimedes (c. 287--212 BC) found an approximation for the circumference of a circle of a given radius.

Since we know that the circumference and diameter of any circle are related by the formula , this means that if we start with a circle of diameter 1, then Archimedes' approximation for actually provides an approximation for .

Archimedes' idea was to approximate the circle using both inscribed and circumscribed (regular) polygons. Below are pictured inscribed and circumscribed octagons.

More generally, we would consider inscribed and circumscribed  -gons. The inscribed  -gon has sides, each of the same length , and the circumscribed  -gon has sides, each of the same length . (In truth, we should consider  -gons, where M is a positive integer. But, for simplicity, forego this extra generality.)

The perimeter of the inscribed  -gon, which we denote by , and the perimeter of the circumscribed  -gon, which we denote by , are approximations for and so, in this case, are also approximations for :

By means of geometric (and what we would now call trigonometric) arguments, Archimedes was able to derive iterative formulas for and , which are reminiscent of the Babylonian algorithm for computing square roots.