Successive application of Archimedes' recurrence formula gives the Archimedes algorithm, which can be used to provide
successive approximations to 
(pi). The algorithm is also called
the Borchardt-Pfaff algorithm. Archimedes obtained the first rigorous approximation
of 
by circumscribing and inscribing 
-gons on a circle.
From Archimedes' recurrence formula,
the circumferences 
and 
of the circumscribed and inscribed polygons
are
 |  |  |
(1)
|
 |  |  |
(2)
|
where
 |
(3)
|
For a hexagon, 
and
 |  |  |
(4)
|
 |  |  |
(5)
|
where 
.
The first iteration of Archimedes' recurrence
formula then gives
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
Additional iterations do not have simple closed forms, but the numerical approximations for 
,
1, 2, 3, 4 (corresponding to 6-, 12-, 24-, 48-, and 96-gons) are
 |
(9)
|
 |
(10)
|
 |
(11)
|
 |
(12)
|
 |
(13)
|
By taking 
(a 96-gon) and using strict inequalities to convert irrational bounds to rational
bounds at each step, Archimedes obtained the slightly looser result
 |
(14)
|
