I just read a few of recent responces. An interesting thing is that nobody mentioned things that are really used for calculating pi, such as arctan or other hypergeometric series, or Simon Plouffe's formula etc. In addition to Richardson's method that I briefly described at the beginning of the thread, another not very often used approach is to use better than sin(x)<x<tan(x) inequalities. For example, sin(x)*(14+cos(x))/(9+6*cos(x)) < x < 8*sin(x)/9/(1+cos(x))+25*sin(x)/(36+9*cos(x)) Both inequalities can be certainly improved, and Richardson's method applied to them would give even better results, but even the cited inequalities give for x = pi/6 3.14156< pi < 3.14161 and for x=pi/12, 3.141592< pi < 3.141593 Compare that with Archimedes calculations; 96*sin(pi/96) < pi < 96*tan (pi/96) requires a lot of work, but gives only 3.141 < pi < 3.1427 Alec Mihailovs