223/71 < π < 22/7
The approximation πa = 22/7 is referred to as
Archimedes Approximation and is very good.
It has been reported that a 2000 B.C. Babylonian approximation is
πb = 25/8. We will compare these two approximations.
The author, in the spirit of idiot's advocate, will venture his
own approximation of πc = 19/6. The
Babylonian approximation is good to one part in 189,
the author's, one part in 125,
and Archimedes an astonishing one part in 2484.
Archimedes' approach is to circumscribe and inscribe regular n-gons around a unit circle. Let Pn be the semi-perimeter of a regular n-gon circumscribed around a unit circle. Let pn be the semi-perimeter of a regular n-gon inscribed in a unit circle. Then,
Pn = n (tan π/n)
pn = n (sin π/n)
and with increasing n the approximations become very close,
pn < π < Pn
Archimedes derives a cotangent half-angle formula
and a cosecant identity,
cot x/2 = cot x + csc x,
csc2 x = 1 + cot2 x.
From the geometry of hexagons we have,
P6 = 6 / Sqrt(3) p6 = 3Iterated applications of the cotangent and cosecant identies, we calculate values for Pn and pn for n = 6 × 2i. Archimedes calculated to n = 96.
Manipulating the formulas, the iteration can be expressed entirely in a semi-perimeter,
P2n = 2 Pn / (1 + Sqrt(1 + (Pn/n)2)))The inscribed semiperimeter is given by a form of the cosecant identity,
(1/pn)2 - (1/Pn)2 = 1/n2.
This formula is shown in a way which gives insight into the convergence
of the inner and outer approximations around π.
However, Archimedes calculated for two separate iterations. To calculate, he need to extract square roots approximately, and to bias the approximation according to the direction to which his approximations were to approach π.
For example, P96 = 3.14271.