# Trigonometry/Calculating Pi

Various formulae for calculating pi can be obtained from the power series expansion for .

Since , we have

This formula (due to Gottfried Leibniz) converges too slowly to be of practical use. However, similar formulae with much faster convergence can be found. John Machin (1680-1752) showed that

- .

This formula was widely used by hand calculators. The first part of the right hand side is easy to calculate since finding **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \frac{1}{5^n}}**
involves very simple division, and the second part only needs 50 terms to compute 240 decimal places.

Leonhard Euler (1707-1783) showed that

- .

Störmer showed that

- ,

and this formula was used in 1962 to calculate to over 100,000 decimals.