The Number Pi

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Pi Formulas

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
An exact formula for  in terms of the inverse tangents of unit fractions is Machin's formula
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
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