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)

Using the identity
 (8)

then gives
 (9)

The second follows from
 (10)

Using the identity
 (11)

gives
 (12)
Successive application gives the Archimedes Algorithm, which can be used to provide successive approximations to Pi ().