There are many formulas of of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations. is intimately related to the properties of circles and spheres. For a circle of radius, the circumference and area are given by
Similarly, for a sphere of radius, the surface area and volume enclosed are
There are three other Machin-like formulas, as well as thousands of other similar formulas having terms.
Gregory and Leibniz found
(Wells 1986, p. 50), which is known as the Gregory series and may be obtained by plugging into the Leibniz series for . The error after the th term of this series in the Gregory series is larger than so this sum converges so slowly that 300 terms are not sufficient to calculate correctly to two decimal places! However, it can be transformed to
of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations.
is intimately related to the properties of circles and spheres. For a circle of radius 
, the circumference and area are given by
, the surface area and volume enclosed are
in terms of the inverse tangents of unit fractions is Machin's formula
Gregory and Leibniz found
into the Leibniz series for 
. The error after the 
th term of this series in the Gregory series is larger than 
so this sum converges so slowly that 300 terms are not sufficient to calculate 
correctly to two decimal places! However, it can be transformed to
The constant pi, denoted 
, is a real number defined as the ratio of a circle's circumference 
to its diameter 
,
has decimal expansion given by
