List of formulae involving π

Part of a series of articles on the
mathematical constant π
3.1415926535897932384626433...
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 Srinivasa Ramanujan John Wrench Chudnovsky brothers Yasumasa Kanada
History
Chronology Book
In culture
Legislation Pi Day
Related topics
Squaring the circle Basel problem Six nines in π Other topics related to π
v t

The following is a list of significant formulae involving the mathematical constant π. Many of these formulae can be found in the article Pi, or the article Approximations of π.

Euclidean geometry[]

${\displaystyle \pi ={\frac {C}{d}}}$

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

${\displaystyle \pi ={\frac {L}{w}}}$

where L and w are, respectively, the perimeter and the width of any curve of constant width.

${\displaystyle A=\pi r^{2}}$

where A is the area of a circle and r is the radius. More generally,

${\displaystyle A=\pi ab}$

where A is the area enclosed by an ellipse with semi-major axis a and semi-minor axis b.

${\displaystyle A=4\pi r^{2}}$

where A is the area between the witch of Agnesi and its asymptotic line; r is the radius of the defining circle.

${\displaystyle A={\frac {\Gamma (1/4)^{2}}{2{\sqrt {\pi }}}}r^{2}={\frac {\pi r^{2}}{\operatorname {agm} (1,1/{\sqrt {2}})}}}$

where A is the area of a squircle with minor radius r, ${\displaystyle \Gamma }$ is the gamma function and ${\displaystyle \operatorname {agm} }$ is the arithmetic–geometric mean.

${\displaystyle A=(k+1)(k+2)\pi r^{2}}$

where A is the area of an epicycloid with the smaller circle of radius r and the larger circle of radius kr (${\displaystyle k\in \mathbb {N} }$), assuming the initial point lies on the larger circle.

${\displaystyle A={\frac {(-1)^{k}+3}{8}}\pi a^{2}}$

where A is the area of a rose with angular frequency k (${\displaystyle k\in \mathbb {N} }$) and amplitude a.

${\displaystyle L={\frac {\Gamma (1/4)^{2}}{\sqrt {\pi }}}c={\frac {2\pi c}{\operatorname {agm} (1,1/{\sqrt {2}})}}}$

where L is the perimeter of the lemniscate of Bernoulli with focal distance c.

${\displaystyle V={4 \over 3}\pi r^{3}}$

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

${\displaystyle SA=4\pi r^{2}}$

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

${\displaystyle H={1 \over 2}\pi ^{2}r^{4}}$

where H is the hypervolume of a 3-sphere and r is the radius.

${\displaystyle SV=2\pi ^{2}r^{3}}$

where SV is the surface volume of a 3-sphere and r is the radius.

Regular convex polygons[]

Sum S of internal angles of a regular convex polygon with n sides:

${\displaystyle S=(n-2)\pi }$

Area A of a regular convex polygon with n sides and side length s:

${\displaystyle A={\frac {ns^{2}}{4}}\cot {\frac {\pi }{n}}}$

Inradius r of a regular convex polygon with n sides and side length s:

${\displaystyle r={\frac {s}{2}}\cot {\frac {\pi }{n}}}$

Circumradius R of a regular convex polygon with n sides and side length s:

${\displaystyle R={\frac {s}{2}}\csc {\frac {\pi }{n}}}$

Physics[]

• The cosmological constant:

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

• Heisenberg's uncertainty principle:

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

• Einstein's field equation of general relativity:

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

• Coulomb's law for the electric force in vacuum:

${\displaystyle F={\frac {|q_{1}q_{2}|}{4\pi \varepsilon _{0}r^{2}}}}$

• Magnetic permeability of free space:

${\displaystyle \mu _{0}\approx 4\pi \cdot 10^{-7}\,\mathrm {N} /\mathrm {A} ^{2}}$

• Period of a simple pendulum with small amplitude:

${\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}}$

• Kepler's third law of planetary motion:

${\displaystyle {\frac {R^{3}}{T^{2}}}={\frac {GM}{4\pi ^{2}}}}$

• The buckling formula:

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

Formulae yielding π[]

Integrals[]

${\displaystyle 2\int _{-1}^{1}{\sqrt {1-x^{2}}}\,dx=\pi }$ (integrating two halves ${\displaystyle y(x)={\sqrt {1-x^{2}}}}$ to obtain the area of the unit circle)

${\displaystyle \int _{-\infty }^{\infty }\operatorname {sech} x\,dx=\pi }$

${\displaystyle \int _{-\infty }^{\infty }\int _{t}^{\infty }e^{-1/2t^{2}-x^{2}+xt}\,dx\,dt=\int _{-\infty }^{\infty }\int _{t}^{\infty }e^{-t^{2}-1/2x^{2}+xt}\,dx\,dt=\pi }$

${\displaystyle \int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}=\pi }$

${\displaystyle \int _{-\infty }^{\infty }{\frac {dx}{1+x^{2}}}=\pi }$^{[1][note 1]} (see also Cauchy distribution)

${\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}}$ (see Gaussian integral).

${\displaystyle \oint {\frac {dz}{z}}=2\pi i}$ (when the path of integration winds once counterclockwise around 0. See also Cauchy's integral formula).

${\displaystyle \int _{0}^{\infty }\ln \left(1+{\frac {1}{x^{2}}}\right)\,dx=\pi }$^[2]

${\displaystyle \int _{-\infty }^{\infty }{\frac {\sin x}{x}}\,dx=\pi }$

${\displaystyle \int _{0}^{1}{x^{4}(1-x)^{4} \over 1+x^{2}}\,dx={22 \over 7}-\pi }$ (see also Proof that 22/7 exceeds π).

${\displaystyle \int _{0}^{\infty }{\frac {x^{\alpha -1}}{x+1}}\,dx={\frac {\pi }{\sin \pi \alpha }},\quad 0<\alpha <1}$

${\displaystyle \int _{0}^{\infty }{\frac {dx}{\sqrt {x(x+a)(x+b)}}}={\frac {\pi }{\operatorname {agm} ({\sqrt {a}},{\sqrt {b}})}}}$ (where ${\displaystyle \operatorname {agm} }$ is the arithmetic–geometric mean;^[3] see also elliptic integral)

Note that with symmetric integrands ${\displaystyle f(-x)=f(x)}$, formulas of the form ${\textstyle \int _{-a}^{a}f(x)\,dx}$ can also be translated to formulas ${\textstyle 2\int _{0}^{a}f(x)\,dx}$.

Efficient infinite series[]

${\displaystyle \sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=\sum _{k=0}^{\infty }{\frac {2^{k}k!^{2}}{(2k+1)!}}={\frac {\pi }{2}}}$ (see also Double factorial)

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

${\displaystyle {\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}={\frac {1}{\pi }}}$ (see Srinivasa Ramanujan, Ramanujan–Sato series)

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

${\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{4^{k}}}\left({\frac {2}{4k+1}}+{\frac {2}{4k+2}}+{\frac {1}{4k+3}}\right)=\pi }$^[4]

${\displaystyle \sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right)=\pi }$ (see Bailey–Borwein–Plouffe formula)

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

Plouffe's series for calculating arbitrary decimal digits of π:^[5]

${\displaystyle \sum _{k=1}^{\infty }k{\frac {2^{k}k!^{2}}{(2k)!}}=\pi +3}$

Other infinite series[]

${\displaystyle \zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}}$ (see also Basel problem and Riemann zeta function)

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

${\displaystyle \zeta (2n)=\sum _{k=1}^{\infty }{\frac {1}{k^{2n}}}\,={\frac {1}{1^{2n}}}+{\frac {1}{2^{2n}}}+{\frac {1}{3^{2n}}}+{\frac {1}{4^{2n}}}+\cdots =(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}}$ , where B_2n is a Bernoulli number.

${\displaystyle \sum _{n=1}^{\infty }{\frac {3^{n}-1}{4^{n}}}\,\zeta (n+1)=\pi }$^[6]

${\displaystyle \sum _{n=2}^{\infty }{\frac {2(3/2)^{n}-3}{n}}(\zeta (n)-1)=\ln \pi }$

${\displaystyle \sum _{n=1}^{\infty }\zeta (2n){\frac {x^{2n}}{n}}=\ln {\frac {\pi x}{\sin \pi x}},\quad 0<|x|<1}$

${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots =\arctan {1}={\frac {\pi }{4}}}$ (see Leibniz formula for pi)

${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{(n^{2}-n)/2}}{2n+1}}=1+{\frac {1}{3}}-{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}+{\frac {1}{11}}-\cdots ={\frac {\pi }{2{\sqrt {2}}}}}$ (Newton, Second Letter to Oldenburg, 1676)

${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{3^{n}(2n+1)}}=1-{\frac {1}{3^{1}\cdot 3}}+{\frac {1}{3^{2}\cdot 5}}-{\frac {1}{3^{3}\cdot 7}}+{\frac {1}{3^{4}\cdot 9}}-\cdots ={\sqrt {3}}\arctan {\frac {1}{\sqrt {3}}}={\frac {\pi }{2{\sqrt {3}}}}}$ (Madhava series)

${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}-{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}-{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{12}}}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{(2n)^{2}}}={\frac {1}{2^{2}}}+{\frac {1}{4^{2}}}+{\frac {1}{6^{2}}}+{\frac {1}{8^{2}}}+\cdots ={\frac {\pi ^{2}}{24}}}$

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

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

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

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

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

${\displaystyle \sum _{n=0}^{\infty }{\binom {\frac {1}{2}}{n}}{\frac {(-1)^{n}}{2n+1}}=1-{\frac {1}{6}}-{\frac {1}{40}}-\cdots ={\frac {\pi }{4}}}$

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{(4n+1)(4n+3)}}={\frac {1}{1\cdot 3}}+{\frac {1}{5\cdot 7}}+{\frac {1}{9\cdot 11}}+\cdots ={\frac {\pi }{8}}}$

${\displaystyle \sum _{n=1}^{\infty }(-1)^{(n^{2}+n)/2+1}\left|G_{\left((-1)^{n+1}+6n-3\right)/4}\right|=|G_{1}|+|G_{2}|-|G_{4}|-|G_{5}|+|G_{7}|+|G_{8}|-|G_{10}|-|G_{11}|+\cdots ={\frac {\sqrt {3}}{\pi }}}$ (see Gregory coefficients)

${\displaystyle \sum _{n=0}^{\infty }{\frac {(1/2)_{n}^{2}}{2^{n}n!^{2}}}\sum _{n=0}^{\infty }{\frac {n(1/2)_{n}^{2}}{2^{n}n!^{2}}}={\frac {1}{\pi }}}$ (where ${\displaystyle (x)_{n}}$ is the rising factorial)^[7]

${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n(n+1)(2n+1)}}=\pi -3}$ (Nilakantha series)

${\displaystyle \sum _{n=1}^{\infty }{\frac {F_{2n}}{n^{2}{\binom {2n}{n}}}}={\frac {4\pi ^{2}}{25{\sqrt {5}}}}}$ (where ${\displaystyle F_{n}}$ is the n-th Fibonacci number)

${\displaystyle \pi =\sum _{n=1}^{\infty }{\frac {(-1)^{\varepsilon (n)}}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}-{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{9}}-{\frac {1}{10}}+{\frac {1}{11}}+{\frac {1}{12}}-{\frac {1}{13}}+\cdots }$   (where ${\displaystyle \varepsilon (n)}$ is the number of prime factors of the form ${\displaystyle p\equiv 1\,(\mathrm {mod} \,4)}$ of ${\displaystyle n}$; Euler, 1748)^[8]

${\displaystyle \pi =\sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{n+1/2}}}$

${\displaystyle \pi ^{2}=\sum _{n=-\infty }^{\infty }{\frac {1}{(n+1/2)^{2}}}}$^[9]

The last two formulas are special cases of

${\displaystyle {\begin{aligned}{\frac {\pi }{\sin \pi x}}&=\sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{n+x}}\\\left({\frac {\pi }{\sin \pi x}}\right)^{2}&=\sum _{n=-\infty }^{\infty }{\frac {1}{(n+x)^{2}}}\end{aligned}}}$

which generate infinitely many analogous formulas for ${\displaystyle \pi }$ when ${\displaystyle x\in \mathbb {Q} \setminus \mathbb {Z} .}$

Some formulas relating π and harmonic numbers are given here. Further infinite series involving π are:^[10]

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

where ${\displaystyle (x)_{n}}$ is the Pochhammer symbol for the rising factorial. See also Ramanujan–Sato series.

Machin-like formulae[]

${\displaystyle {\frac {\pi }{4}}=\arctan 1}$

${\displaystyle {\frac {\pi }{4}}=\arctan {\frac {1}{2}}+\arctan {\frac {1}{3}}}$

${\displaystyle {\frac {\pi }{4}}=2\arctan {\frac {1}{2}}-\arctan {\frac {1}{7}}}$

${\displaystyle {\frac {\pi }{4}}=2\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}}$

${\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}}$ (the original Machin's formula)

${\displaystyle {\frac {\pi }{4}}=5\arctan {\frac {1}{7}}+2\arctan {\frac {3}{79}}}$

${\displaystyle {\frac {\pi }{4}}=6\arctan {\frac {1}{8}}+2\arctan {\frac {1}{57}}+\arctan {\frac {1}{239}}}$

${\displaystyle {\frac {\pi }{4}}=12\arctan {\frac {1}{49}}+32\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}+12\arctan {\frac {1}{110443}}}$

${\displaystyle {\frac {\pi }{4}}=44\arctan {\frac {1}{57}}+7\arctan {\frac {1}{239}}-12\arctan {\frac {1}{682}}+24\arctan {\frac {1}{12943}}}$

Infinite products[]

${\displaystyle {\frac {\pi }{4}}=\left(\prod _{p\equiv 1{\pmod {4}}}{\frac {p}{p-1}}\right)\cdot \left(\prod _{p\equiv 3{\pmod {4}}}{\frac {p}{p+1}}\right)={\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {7}{8}}\cdot {\frac {11}{12}}\cdot {\frac {13}{12}}\cdots ,}$ (Euler)

where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator.

${\displaystyle {\frac {{\sqrt {3}}\pi }{6}}=\left(\displaystyle \prod _{p\equiv 1{\pmod {6}} \atop p\in \mathbb {P} }{\frac {p}{p-1}}\right)\cdot \left(\displaystyle \prod _{p\equiv 5{\pmod {6}} \atop p\in \mathbb {P} }{\frac {p}{p+1}}\right)={\frac {5}{6}}\cdot {\frac {7}{6}}\cdot {\frac {11}{12}}\cdot {\frac {13}{12}}\cdot {\frac {17}{18}}\cdots ,}$^{[citation needed]}

${\displaystyle {\frac {\pi }{2}}=\prod _{n=1}^{\infty }{\frac {(2n)(2n)}{(2n-1)(2n+1)}}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots }$ (see also Wallis product)

${\displaystyle {\frac {\pi }{2}}=\prod _{n=1}^{\infty }\left(1+{\frac {1}{n}}\right)^{(-1)^{n+1}}=\left(1+{\frac {1}{1}}\right)^{+1}\left(1+{\frac {1}{2}}\right)^{-1}\left(1+{\frac {1}{3}}\right)^{+1}\cdots }$ (another form of Wallis product)

Viète's formula:

${\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdot \cdots }$

A double infinite product formula involving the Thue–Morse sequence:

${\displaystyle {\frac {\pi }{2}}=\prod _{m\geq 1}\prod _{n\geq 1}\left({\frac {(4m^{2}+n-2)(4m^{2}+2n-1)^{2}}{4(2m^{2}+n-1)(4m^{2}+n-1)(2m^{2}+n)}}\right)^{\epsilon _{n}},}$

where ${\displaystyle \epsilon _{n}=(-1)^{t_{n}}}$ and ${\displaystyle t_{n}}$ is the Thue–Morse sequence (Tóth 2020).

Arctangent formulas[]

${\displaystyle {\frac {\pi }{2^{k+1}}}=\arctan {\frac {\sqrt {2-a_{k-1}}}{a_{k}}},\qquad \qquad k\geq 2}$

${\displaystyle {\frac {\pi }{4}}=\sum _{k\geq 2}\arctan {\frac {\sqrt {2-a_{k-1}}}{a_{k}}},}$

where ${\displaystyle a_{k}={\sqrt {2+a_{k-1}}}}$ such that ${\displaystyle a_{1}={\sqrt {2}}}$.

${\displaystyle {\frac {\pi }{2}}=\sum _{k=0}^{\infty }\arctan {\frac {1}{F_{2k+1}}}=\arctan {\frac {1}{1}}+\arctan {\frac {1}{2}}+\arctan {\frac {1}{5}}+\arctan {\frac {1}{13}}+\cdots }$

where ${\displaystyle F_{k}}$ is the k-th Fibonacci number.

${\displaystyle \pi =\arctan a+\arctan b+\arctan c}$

whenever ${\displaystyle a+b+c=abc}$ and ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ are positive real numbers (see List of trigonometric identities). A special case is

${\displaystyle \pi =\arctan 1+\arctan 2+\arctan 3.}$

Complex exponential formulas[]

${\displaystyle e^{i\pi }+1=0}$ (Euler's identity)

The following equivalences are true for any complex ${\displaystyle z}$:

${\displaystyle e^{z}\in \mathbb {R} \leftrightarrow \Im z\in \pi \mathbb {Z} }$

${\displaystyle e^{z}=1\leftrightarrow z\in 2\pi i\mathbb {Z} }$

Continued fractions[]

${\displaystyle \pi ={3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+{\cfrac {7^{2}}{6+\ddots \,}}}}}}}}}}$

${\displaystyle \pi ={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+{\cfrac {4^{2}}{9+\ddots }}}}}}}}}}}$

${\displaystyle \pi ={\cfrac {4}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+{\cfrac {7^{2}}{2+\ddots }}}}}}}}}}}$

${\displaystyle 2\pi ={6+{\cfrac {2^{2}}{12+{\cfrac {6^{2}}{12+{\cfrac {10^{2}}{12+{\cfrac {14^{2}}{12+{\cfrac {18^{2}}{12+\ddots }}}}}}}}}}}}$

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

(See also Continued fraction and Generalized continued fraction.)

Iterative algorithms[]

${\displaystyle a_{0}=1,\,a_{n+1}=\left(1+{\frac {1}{2n+1}}\right)a_{n},\,\pi =\lim _{n\to \infty }{\frac {a_{n}^{2}}{n}}}$

${\displaystyle a_{1}=0,\,a_{n+1}={\sqrt {2+a_{n}}},\,\pi =\lim _{n\to \infty }2^{n}{\sqrt {2-a_{n}}}}$ (closely related to Viète's formula)

${\displaystyle \omega (i_{n},i_{n-1},\dots ,i_{1})=2+i_{n}{\sqrt {2+i_{n-1}{\sqrt {2+\cdots +i_{1}{\sqrt {2}}}}}}=\omega (b_{n},b_{n-1},\dots ,b_{1}),\,i_{k}\in \{-1,1\},\,b_{k}={\begin{cases}0&{\text{if }}i_{k}=1\\1&{\text{if }}i_{k}=-1\end{cases}},\,\pi ={\displaystyle \lim _{n\rightarrow \infty }{\frac {2^{n+1}}{2h+1}}{\sqrt {\omega \left(\underbrace {10\ldots 0} _{n-m}g_{m,h+1}\right)}}}}$ (where ${\displaystyle g_{m,h+1}}$ is the h+1-th entry of m-bit Gray code, ${\displaystyle h\in \left\{0,1,\ldots ,2^{m}-1\right\}}$)^[11]

${\displaystyle a_{1}=1,\,a_{n+1}=a_{n}+\sin a_{n},\,\pi =\lim _{n\to \infty }a_{n}}$ (cubic convergence)^[12]

${\displaystyle a_{0}=2{\sqrt {3}},\,b_{0}=3,\,a_{n+1}=\operatorname {hm} (a_{n},b_{n}),\,b_{n+1}=\operatorname {gm} (a_{n+1},b_{n}),\,\pi =\lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}}$ (Archimedes' algorithm, see also harmonic mean and geometric mean)^[13]

For more iterative algorithms, see the Gauss–Legendre algorithm and Borwein's algorithm.

Miscellaneous[]

${\displaystyle \Gamma (s)\Gamma (1-s)={\frac {\pi }{\sin \pi s}}}$ (Euler's reflection formula, see Gamma function)

${\displaystyle \pi ^{-s/2}\Gamma \left({\frac {s}{2}}\right)\zeta (s)=\pi ^{-(1-s)/2}\Gamma \left({\frac {1-s}{2}}\right)\zeta (1-s)}$ (the functional equation of the Riemann zeta function)

${\displaystyle e^{-\zeta '(0)}={\sqrt {2\pi }}}$

${\displaystyle e^{\zeta '(0,1/2)-\zeta '(0,1)}={\sqrt {\pi }}}$ (where ${\displaystyle \zeta (s,a)}$ is the Hurwitz zeta function and the derivative is taken with respect to the first variable)

${\displaystyle {\binom {2n}{n}}\sim {\frac {4^{n}}{\sqrt {\pi n}}}}$ (asymptotic growth rate of the central binomial coefficients)

${\displaystyle C_{n}\sim {\frac {4^{n}}{\sqrt {\pi n^{3}}}}}$ (asymptotic growth rate of the Catalan numbers)

${\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}$ (Stirling's approximation)

${\displaystyle \sum _{k=1}^{n}\varphi (k)\sim {\frac {3n^{2}}{\pi ^{2}}}}$ (where ${\displaystyle \varphi }$ is Euler's totient function)

${\displaystyle \sum _{k=1}^{n}{\frac {\varphi (k)}{k}}\sim {\frac {6n}{\pi ^{2}}}}$

${\displaystyle \pi =\mathrm {B} (1/2,1/2)=\Gamma (1/2)^{2}}$ (see also Beta function)

${\displaystyle \pi ={\frac {\Gamma (3/4)^{4}}{\operatorname {agm} (1,1/{\sqrt {2}})^{2}}}={\frac {\Gamma \left({1/4}\right)^{4/3}\operatorname {agm} (1,{\sqrt {2}})^{2/3}}{2}}}$ (where agm is the arithmetic–geometric mean)

${\displaystyle \pi =\operatorname {agm} \left(\theta _{2}^{2}(1/e),\theta _{3}^{2}(1/e)\right)}$ (where ${\displaystyle \theta _{2}}$ and ${\displaystyle \theta _{3}}$ are the Jacobi theta functions^[14])

${\displaystyle \pi =-{\frac {\operatorname {K} (k)}{\operatorname {K} \left({\sqrt {1-k^{2}}}\right)}}\ln q,\quad k={\frac {\theta _{2}^{2}(q)}{\theta _{3}^{2}(q)}}}$ (where ${\displaystyle q\in (0,1)}$ and ${\displaystyle \operatorname {K} (k)}$ is the complete elliptic integral of the first kind with modulus ${\displaystyle k}$; reflecting the nome-modulus inversion problem)^[15]

${\displaystyle \pi =-{\frac {\operatorname {agm} \left(1,{\sqrt {1-k'^{2}}}\right)}{\operatorname {agm} (1,k')}}\ln q,\quad k'={\frac {\theta _{4}^{2}(q)}{\theta _{3}^{2}(q)}}}$ (where ${\displaystyle q\in (0,1)}$)^[15]

${\displaystyle \operatorname {agm} (1,{\sqrt {2}})={\frac {\pi }{\varpi }}}$ (due to Gauss,^[16] ${\displaystyle \varpi }$ is the lemniscate constant)

${\displaystyle i\pi =\operatorname {Log} (-1)=\lim _{n\to \infty }n\left((-1)^{1/n}-1\right)}$ (where ${\displaystyle \operatorname {Log} }$ is the principal value of the complex logarithm)^{[note 2]}

${\displaystyle 1-{\frac {\pi ^{2}}{12}}=\lim _{n\rightarrow \infty }{\frac {1}{n^{2}}}\sum _{k=1}^{n}(n{\bmod {k}})}$ (where ${\textstyle n{\bmod {k}}}$ is the remainder upon division of n by k)

${\displaystyle \pi =\lim _{r\to \infty }{\frac {1}{r^{2}}}\sum _{x=-r}^{r}\;\sum _{y=-r}^{r}{\begin{cases}1&{\text{if }}{\sqrt {x^{2}+y^{2}}}\leq r\\0&{\text{if }}{\sqrt {x^{2}+y^{2}}}>r\end{cases}}}$ (summing a circle's area)

${\displaystyle \pi =\lim _{n\rightarrow \infty }{\frac {4}{n^{2}}}\sum _{k=1}^{n}{\sqrt {n^{2}-k^{2}}}}$ (Riemann sum to evaluate the area of the unit circle)

${\displaystyle \pi =\lim _{n\to \infty }{\frac {2^{4n}n!^{4}}{n(2n)!^{2}}}=\lim _{n\rightarrow \infty }{\frac {2^{4n}}{n{2n \choose n}^{2}}}=\lim _{n\rightarrow \infty }{\frac {1}{n}}\left({\frac {(2n)!!}{(2n-1)!!}}\right)^{2}}$ (by combining Stirling's approximation with Wallis product)

${\displaystyle \pi =\lim _{n\to \infty }{\frac {1}{n}}\ln {\frac {16}{\lambda (ni)}}}$ (where ${\displaystyle \lambda }$ is the modular lambda function)^{[17][note 3]}

${\displaystyle \pi =\lim _{n\to \infty }{\frac {24}{\sqrt {n}}}\ln \left(2^{1/4}G_{n}\right)=\lim _{n\to \infty }{\frac {24}{\sqrt {n}}}\ln \left(2^{1/4}g_{n}\right)}$ (where ${\displaystyle G_{n}}$ and ${\displaystyle g_{n}}$ are Ramanujan's class invariants)^{[18][note 4]}

See also[]

• List of mathematical identities
• Lists of mathematics topics – Wikipedia list article
• List of trigonometric identities – Equalities that involve trigonometric functions
• List of topics related to π – Topics related to the mathematical constant
• List of representations of e

References[]

Notes[]

Other[]

• Tóth, László (2020), "Transcendental Infinite Products Associated with the +-1 Thue-Morse Sequence" (PDF), Journal of Integer Sequences, 23: 20.8.2, arXiv:2009.02025.

Further reading[]

• Peter Borwein, The Amazing Number Pi
• Kazuya Kato, Nobushige Kurokawa, Saito Takeshi: Number Theory 1: Fermat's Dream. American Mathematical Society, Providence 1993, ISBN 0-8218-0863-X.