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 (
).
