List of formulae involving π

mathematical constant $π$
Part of a series of articles on the

Uses
Area of a circle Circumference Use in other formulae
Properties
Irrationality Transcendence
Value
Less than 22/7 Approximations Memorization
People
Archimedes Liu Hui Zu Chongzhi Aryabhata Madhava Ludolph van Ceulen Seki Takakazu Takebe Kenko William Jones John Machin William Shanks John Wrench Chudnovsky brothers Yasumasa Kanada
History
Chronology Book
In culture
Legislation Holiday
Related topics
Squaring the circle Basel problem Six nines in $π$ Other topics related to $π$

The following is a list of significant formulae involving the mathematical constant $π$ . 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 $π$ .

Euclidean Geometry

\pi =C/d\!

where $C$ is the circumference of a circle, $d$ is the diameter.

A=\pi r^{2}\!

where $A$ is the area of a circle and $r$ is the radius.

V={4 \over 3}\pi r^{3}\!

where $V$ is the volume of a sphere and $r$ is the radius.

SA=4\pi r^{2}\!

where $SA$ is the surface area of a sphere and $r$ is the radius.

Physics

The cosmological constant:

\Lambda ={{8\pi G} \over {3c^{2}}}\rho \!

Heisenberg's uncertainty principle:

\Delta x\,\Delta p\geq {\frac {h}{4\pi }}\!

Einstein's field equation of general relativity:

R_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }R+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }\!

Coulomb's law for the electric force:

F={\frac {\left|q_{1}q_{2}\right|}{4\pi \varepsilon _{0}r^{2}}}\!

Magnetic permeability of free space:

\mu _{0}=4\pi \cdot 10^{-7}\,\mathrm {N/A^{2}} \!

Period of a simple pendulum with small amplitude:

T\approx 2\pi {\sqrt {\frac {L}{g}}}\!

The buckling formula:

F={\frac {\pi ^{2}EI}{L^{2}}}

Formulae yielding $π$

Integrals

\int \limits _{-\infty }^{\infty }{\text{sech}}(x)dx=\pi \!

\int \limits _{-\infty }^{\infty }\int \limits _{t}^{\infty }e^{-1/2t^{2}-x^{2}+xt}dxdt=\int \limits _{-\infty }^{\infty }\int \limits _{t}^{\infty }e^{-t^{2}-1/2x^{2}+xt}dxdt=\pi \!

\int \limits _{-1}^{1}{\sqrt {1-x^{2}}}\,dx={\frac {\pi }{2}}\!

\int \limits _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}=\pi \!

\int \limits _{-\infty }^{\infty }{\frac {dx}{1+x^{2}}}=\pi \!

(integral form of arctan over its entire domain, giving the period of tan).

\int \limits _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}\!

(see Gaussian integral).

\oint {\frac {dz}{z}}=2\pi i\!

(when the path of integration winds once counterclockwise around 0. See also Cauchy's integral formula)

\int \limits _{-\infty }^{\infty }{\frac {\sin x}{x}}\,dx=\pi \!

\int \limits _{0}^{1}{x^{4}(1-x)^{4} \over 1+x^{2}}\,dx={22 \over 7}-\pi \!

(see also Proof that 22/7 exceeds π).

Efficient infinite series

\sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=\sum _{k=0}^{\infty }{\frac {2^{k}k!^{2}}{(2k+1)!}}={\frac {\pi }{2}}\!

(see also Double factorial)

12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}={\frac {1}{\pi }}\!

(see Chudnovsky algorithm)

{\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}={\frac {1}{\pi }}\!

(see Srinivasa Ramanujan, Ramanujan–Sato series)

{\frac {\sqrt {3}}{6^{5}}}\sum _{k=0}^{\infty }{\frac {((4k)!)^{2}(6k)!}{9^{k+1}(12k)!(2k)!}}\left({\frac {127169}{12k+1}}-{\frac {1070}{12k+5}}-{\frac {131}{12k+7}}+{\frac {2}{12k+11}}\right)=\pi \!

^[1]

The following are efficient for calculating arbitrary binary digits of $π$ :

\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right)=\pi \!

(see Bailey–Borwein–Plouffe formula)

{\frac {1}{2^{6}}}\sum _{n=0}^{\infty }{\frac {{(-1)}^{n}}{2^{10n}}}\left(-{\frac {2^{5}}{4n+1}}-{\frac {1}{4n+3}}+{\frac {2^{8}}{10n+1}}-{\frac {2^{6}}{10n+3}}-{\frac {2^{2}}{10n+5}}-{\frac {2^{2}}{10n+7}}+{\frac {1}{10n+9}}\right)=\pi \!

Other infinite series

\zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}\!

(see also Basel problem and Riemann zeta function)

\zeta (4)={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}\!

\zeta (2n)=\sum _{k=1}^{\infty }{\frac {1}{k^{2n}}}\,={\frac {1}{1^{2n}}}+{\frac {1}{2^{2n}}}+{\frac {1}{3^{2n}}}+{\frac {1}{4^{2n}}}+\cdots =(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}\!

, where B_2n is a Bernoulli number.

\sum _{n=1}^{\infty }{\frac {3^{n}-1}{4^{n}}}\,\zeta (n+1)=\pi \!

^[2]

\sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{1}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots =\arctan {1}={\frac {\pi }{4}}\!

(see Leibniz formula for pi)

\sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{2}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots ={\frac {\pi ^{2}}{8}}\!

\sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{3}={\frac {1}{1^{3}}}-{\frac {1}{3^{3}}}+{\frac {1}{5^{3}}}-{\frac {1}{7^{3}}}+\cdots ={\frac {\pi ^{3}}{32}}\!

\sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{4}={\frac {1}{1^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{5^{4}}}+{\frac {1}{7^{4}}}+\cdots ={\frac {\pi ^{4}}{96}}\!

\sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{5}={\frac {1}{1^{5}}}-{\frac {1}{3^{5}}}+{\frac {1}{5^{5}}}-{\frac {1}{7^{5}}}+\cdots ={\frac {5\pi ^{5}}{1536}}\!

\sum _{n=0}^{\infty }{\left({\frac {(-1)^{n}}{2n+1}}\right)}^{6}={\frac {1}{1^{6}}}+{\frac {1}{3^{6}}}+{\frac {1}{5^{6}}}+{\frac {1}{7^{6}}}+\cdots ={\frac {\pi ^{6}}{960}}\!

\pi ={1}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}-{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{9}}-{\frac {1}{10}}+{\frac {1}{11}}+{\frac {1}{12}}-{\frac {1}{13}}+\cdots \!

(Euler, 1748)

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.^[3]

Also:

\sum _{n=1}^{\infty }{\frac {F_{2n}}{n^{2}{\binom {2n}{n}}}}={\frac {4\pi ^{2}}{25{\sqrt {5}}}}

where

F_{n}

is the n-th Fibonacci number.

Machin-like formulae

Infinite series

Some infinite series involving pi are:^[4]

$\pi ={\frac {1}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {((2n)!)^{3}(42n+5)}{(n!)^{6}{16}^{3n+1}}}\!$
$\pi ={\frac {4}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(4n)!(21460n+1123)}{(n!)^{4}{441}^{2n+1}{2}^{10n+1}}}$
$\pi ={\frac {4}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(6n+1)\left({\frac {1}{2}}\right)_{n}^{3}}{{4^{n}}(n!)^{3}}}\!$
$\pi ={\frac {32}{Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {{\sqrt {5}}-1}{2}}\right)^{8n}{\frac {(42n{\sqrt {5}}+30n+5{\sqrt {5}}-1)\left({\frac {1}{2}}\right)_{n}^{3}}{{64^{n}}(n!)^{3}}}\!$
$\pi ={\frac {27}{4Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {2}{27}}\right)^{n}{\frac {(15n+2)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{3}}\right)_{n}\left({\frac {2}{3}}\right)_{n}}{(n!)^{3}}}\!$
$\pi ={\frac {15{\sqrt {3}}}{2Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {4}{125}}\right)^{n}{\frac {(33n+4)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{3}}\right)_{n}\left({\frac {2}{3}}\right)_{n}}{(n!)^{3}}}\!$
$\pi ={\frac {85{\sqrt {85}}}{18{\sqrt {3}}Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {4}{85}}\right)^{n}{\frac {(133n+8)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{6}}\right)_{n}\left({\frac {5}{6}}\right)_{n}}{(n!)^{3}}}\!$
$\pi ={\frac {5{\sqrt {5}}}{2{\sqrt {3}}Z}}\!$	$Z=\sum _{n=0}^{\infty }\left({\frac {4}{125}}\right)^{n}{\frac {(11n+1)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{6}}\right)_{n}\left({\frac {5}{6}}\right)_{n}}{(n!)^{3}}}\!$
$\pi ={\frac {2{\sqrt {3}}}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(8n+1)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{9}^{n}}}\!$
$\pi ={\frac {\sqrt {3}}{9Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(40n+3)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{49}^{2n+1}}}\!$
$\pi ={\frac {2{\sqrt {11}}}{11Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(280n+19)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{99}^{2n+1}}}\!$
$\pi ={\frac {\sqrt {2}}{4Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(10n+1)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{9}^{2n+1}}}\!$
$\pi ={\frac {4{\sqrt {5}}}{5Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(644n+41)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}5^{n}{72}^{2n+1}}}\!$
$\pi ={\frac {4{\sqrt {3}}}{3Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(28n+3)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{3^{n}}{4}^{n+1}}}\!$
$\pi ={\frac {4}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(20n+3)\left({\frac {1}{2}}\right)_{n}\left({\frac {1}{4}}\right)_{n}\left({\frac {3}{4}}\right)_{n}}{(n!)^{3}{2}^{2n+1}}}\!$
$\pi ={\frac {72}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(4n)!(260n+23)}{(n!)^{4}4^{4n}18^{2n}}}\!$
$\pi ={\frac {3528}{Z}}\!$	$Z=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(4n)!(21460n+1123)}{(n!)^{4}4^{4n}882^{2n}}}\!$