Borchardt-Pfaff algorithm

Successive application of Archimedes' recurrence formula gives the Archimedes algorithm, which can be used to provide successive approximations to π (pi). The algorithm is also called the Borchardt-Pfaff algorithm. Archimedes obtained the first rigorous approximation of π by circumscribing and inscribing n = 6·2^k-gons on a circle. From Archimedes' recurrence formula, the circumferences a and b of the circumscribed and inscribed polygons are a(n) | = | 2n tan(π/n) b(n) | = | 2n sin(π/n),