What is formulas to calculate pi?

Answer 1

There are many formulae to calculate pi.

The simplest, geometric one, is the ratio of the circumference of a circle to its diameter.

But there are many infinite series such as:

pi = 4*(1 - 1/3 + 1/5 - 1/7 + 1/9 - ... )

or

pi = sqrt[6*(1/12 + 1/22 + 1/32 + 1/42 + ... )]

or

pi = 3 + 4/(2*3*4) - 4/(4*5*6) + 4/(6*7*8) - 4/(8*9*10) + ...

Some converge to pi quickly, others are very slow. The first infinite series given here (Gregory-Leibniz) requires half a million terms to give pi accurate to 5 decimal digits! The other two are faster. Some, like the Chudnovsky series, below are very fast.

1/pi = 12*[sum from k = 0 to infinity of](-1)k*(6k)!(13591409 + 545140134k)/[(3k)!*(k!)3*6403203k+3/2]

generates 14 correct decimal digits for each additional term. This series was used by Alexander J Yee & Shigeru Kondo to calculate pi to 10 trillion (and 50) digits in October 2011.