Archimedes' Recurrence Formula

Let  and  be the perimeters of the circumscribed and inscribed -gon and  and  the perimeters of the circumscribed and inscribed -gon. Then

			(1)
			(2)

The first follows from the fact that side lengths of the polygons on a circle of radius  are

			(3)
			(4)

so

			(5)
			(6)

But

			(7)
			(8)

Using the identity

(9)

then gives

(10)

The second follows from

(11)

Using the identity

(12)

gives

			(13)
			(14)
			(15)
			(16)

Successive application gives the Archimedes algorithm, which can be used to provide successive approximations to pi ().

REFERENCES:

Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 186, 1965.