List of formulae involving π


The following is a list of significant formulae involving the mathematical constant pi|. The list contains only formulae whose significance is established either in the article on the formula itself, the article Pi, or the article Approximations of.

Euclidean geometry

where is the circumference of a circle, is the diameter.
where is the area of a circle and is the radius.
where is the volume of a sphere and is the radius.
where is the surface area of a sphere and is the radius.

Physics

Integrals

Note that with symmetric integrands, formulas of the form can also be translated to formulas.

Efficient infinite series

The following are efficient for calculating arbitrary binary digits of :

Other infinite series

After the first two terms, the signs are determined as follows: If the denominator is a prime of the form 4m − 1, the sign is positive; if the denominator is a prime of the form 4m + 1, the sign is negative; for composite numbers, the sign is equal the product of the signs of its factors.
Also:
where is the n-th Fibonacci number.
Some formulas relating and harmonic numbers are given here.

Machin-like formulae

where is the n-th Fibonacci number.

Infinite series

Some infinite series involving π are:

where is the Pochhammer symbol for the rising factorial. See also Ramanujan–Sato series.

Infinite products

Viète's formula:

Arctangent formulas

where such that.

Continued fractions

For more on the third identity, see Euler's continued fraction formula.

Miscellaneous