How to calculate π 
Possible Duplicate:
Simple numerical methods for calculating the digits of Pi 

How do people/computers calculate π?
Im sure long ago, someone just took a measurement of the circumference of circles and their radii and came up with 3.  Then as measurements got more accurate 3.14... etc...
But how do modern people, especially computers calculate π to trillions of digits?
If π is an irrational number, then how do we know the calculations are correct?
 A: If you really want the answer then this paper might be helpful
http://www.pnas.org/content/86/21/8178
A: Archimedes considered a 96-sided regular polygon inscribed in a circle and another circumscribed about the circle, and found their perimeters, and thereby showed that $3+\dfrac{10}{71} < \pi < 3+ \dfrac 1 7$.
More recently (the past two or three centuries) algorithms derived from the power series for the arctangent function have been used.
In Wikipedia's List of topics related to $\pi$ I find these:


*

*Chronology of computation of π

*Bailey–Borwein–Plouffe formula

*Borwein's algorithm

*Gauss–Legendre algorithm

*Proof that 22/7 exceeds π

*Leibniz formula for π

*Liu Hui's π algorithm

*Approximations of π
Wikipedia's  List of topics related to pi --- actually a list of Wikipedia articles on those topics --- is quite interesting:


*

*2π theorem

*Approximations of π

*Arithmetic-geometric mean

*Bailey–Borwein–Plouffe formula

*Basel problem

*Borwein's algorithm

*Buffon's needle

*Cadaeic Cadenza

*Chronology of computation of π

*Circle

*Euler's identity

*Feynman point

*Gauss–Legendre algorithm

*Gaussian function

*History of π

*A History of Pi (book)

*Indiana Pi Bill

*Leibniz formula for pi

*Lindemann–Weierstrass theorem (Proof that π is transcendental)

*List of circle topics

*List of formulae involving π

*Liu Hui's π algorithm

*Mathematical constant (sorted by continued fraction representation)

*Method of exhaustion

*Milü

*Pi

*Pi (letter)

*Pi Day

*PiFast

*PiHex

*Pilish

*Pimania (computer game)

*Piphilology

*Proof that π is irrational

*Proof that 22/7 exceeds π

*Proof of Wallis product

*Rabbi Nehemiah

*Radian

*Rhind Mathematical Papyrus

*Salamin–Brent algorithm

*Software for calculating π

*Squaring the circle

*Tau (2π)

*Turn (geometry)

*Viète's formula

