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
π = C d = C 2 r 2 r + 1 {\displaystyle \pi ={\frac {C}{d}}={\frac {C}{2r^{2r+1}}}}
where C is the circumference of a circle, d is the diameter, and r is the radius. More generally,
π = L w {\displaystyle \pi ={\frac {L}{w}}}
where L and w are, respectively, the perimeter and the width of any curve of constant width.
A = π r 2 {\displaystyle A=\pi r^{2}}
where A is the area of a circle. More generally,
A = π a b {\displaystyle A=\pi ab}
where A is the area enclosed by an ellipse with semi-major axis a and semi-minor axis b.
C = 2 π agm ( a , b ) ( a 1 2 − ∑ n = 2 ∞ 2 n − 1 ( a n 2 − b n 2 ) ) {\displaystyle C={\frac {2\pi }{\operatorname {agm} (a,b)}}\left(a_{1}^{2}-\sum _{n=2}^{\infty }2^{n-1}(a_{n}^{2}-b_{n}^{2})\right)}
where C is the circumference of an ellipse with semi-major axis a and semi-minor axis b and a n , b n {\displaystyle a_{n},b_{n}} are the arithmetic and geometric iterations of agm ( a , b ) {\displaystyle \operatorname {agm} (a,b)} , the arithmetic-geometric mean of a and b with the initial values a 0 = a {\displaystyle a_{0}=a} and b 0 = b {\displaystyle b_{0}=b} .
A = 4 π r 2 {\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.
A = Γ ( 1 / 4 ) 2 2 π r 2 = π r 2 agm ( 1 , 1 / 2 ) {\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, Γ {\displaystyle \Gamma } is the gamma function.
A = ( k + 1 ) ( k + 2 ) π r 2 {\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 ( k ∈ N {\displaystyle k\in \mathbb {N} } ), assuming the initial point lies on the larger circle.
A = ( − 1 ) k + 3 8 π a 2 {\displaystyle A={\frac {(-1)^{k}+3}{8}}\pi a^{2}}
where A is the area of a rose with angular frequency k ( k ∈ N {\displaystyle k\in \mathbb {N} } ) and amplitude a.
L = Γ ( 1 / 4 ) 2 π c = 2 π c agm ( 1 , 1 / 2 ) {\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.
V = 4 3 π r 3 {\displaystyle V={4 \over 3}\pi r^{3}}
where V is the volume of a sphere and r is the radius.
S A = 4 π r 2 {\displaystyle SA=4\pi r^{2}}
where SA is the surface area of a sphere and r is the radius.
H V = 1 2 π 2 r 4 {\displaystyle HV={1 \over 2}\pi ^{2}r^{4}}
where HV is the hypervolume of a 3-sphere and r is the radius.
S V = 2 π 2 r 3 {\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:
S = ( n − 2 ) π {\displaystyle S=(n-2)\pi }
Area A of a regular convex polygon with n sides and side length s:
A = n s 2 4 cot π n {\displaystyle A={\frac {ns^{2}}{4}}\cot {\frac {\pi }{n}}}
Inradius r of a regular convex polygon with n sides and side length s:
r = s 2 cot π n {\displaystyle r={\frac {s}{2}}\cot {\frac {\pi }{n}}}
Circumradius R of a regular convex polygon with n sides and side length s:
R = s 2 csc π n {\displaystyle R={\frac {s}{2}}\csc {\frac {\pi }{n}}}
Physics The cosmological constant:
Λ = 8 π G 3 c 2 ρ {\displaystyle \Lambda ={{8\pi G} \over {3c^{2}}}\rho }
Heisenberg's uncertainty principle:
Δ x Δ p ≥ h 4 π {\displaystyle \Delta x\,\Delta p\geq {\frac {h}{4\pi }}}
Einstein's field equation of general relativity:
R μ ν − 1 2 g μ ν R + Λ g μ ν = 8 π G c 4 T μ ν {\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:
F = | q 1 q 2 | 4 π ε 0 r 2 {\displaystyle F={\frac {|q_{1}q_{2}|}{4\pi \varepsilon _{0}r^{2}}}}
Magnetic permeability of free space:
μ 0 ≈ 4 π ⋅ 10 − 7 N / A 2 {\displaystyle \mu _{0}\approx 4\pi \cdot 10^{-7}\,\mathrm {N} /\mathrm {A} ^{2}}
Approximate period of a simple pendulum with small amplitude:
T ≈ 2 π L g {\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}}
Exact period of a simple pendulum with amplitude θ 0 {\displaystyle \theta _{0}} ( agm {\displaystyle \operatorname {agm} } is the arithmetic–geometric mean):
T = 2 π agm ( 1 , cos ( θ 0 / 2 ) ) L g {\displaystyle T={\frac {2\pi }{\operatorname {agm} (1,\cos(\theta _{0}/2))}}{\sqrt {\frac {L}{g}}}}
Period of a spring-mass system with spring constant k {\displaystyle k} and mass m {\displaystyle m} :
T = 2 π m k {\displaystyle T=2\pi {\sqrt {\frac {m}{k}}}}
Kepler's third law of planetary motion:
R 3 T 2 = G M 4 π 2 {\displaystyle {\frac {R^{3}}{T^{2}}}={\frac {GM}{4\pi ^{2}}}}
The buckling formula:
F = π 2 E I L 2 {\displaystyle F={\frac {\pi ^{2}EI}{L^{2}}}}
A puzzle involving "colliding billiard balls":
⌊ b N π ⌋ {\displaystyle \lfloor {b^{N}\pi }\rfloor }
is the number of collisions made (in ideal conditions, perfectly elastic with no friction) by an object of mass m initially at rest between a fixed wall and another object of mass b2Nm, when struck by the other object. (This gives the digits of π in base b up to N digits past the radix point.)
Formulae yielding π
Integrals
2 ∫ − 1 1 1 − x 2 d x = π {\displaystyle 2\int _{-1}^{1}{\sqrt {1-x^{2}}}\,dx=\pi } (integrating two halves y ( x ) = 1 − x 2 {\displaystyle y(x)={\sqrt {1-x^{2}}}} to obtain the area of the unit circle)
∫ 0 2 4 − x 2 d x = π {\displaystyle \int _{0}^{2}{\sqrt {4-x^{2}}}\,dx=\pi } (integrating a quarter of a circle with a radius of two x 2 + y 2 = 4 {\displaystyle x^{2}+y^{2}=4} to obtain 4 π / 4 {\displaystyle {4\pi }/4} )
∫ − ∞ ∞ sech x d x = π {\displaystyle \int _{-\infty }^{\infty }\operatorname {sech} x\,dx=\pi }
∫ − ∞ ∞ ∫ t ∞ e − 1 / 2 t 2 − x 2 + x t d x d t = ∫ − ∞ ∞ ∫ t ∞ e − t 2 − 1 / 2 x 2 + x t d x d t = π {\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 }
∫ − 1 1 d x 1 − x 2 = π {\displaystyle \int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}=\pi }
∫ − ∞ ∞ d x 1 + x 2 = π {\displaystyle \int _{-\infty }^{\infty }{\frac {dx}{1+x^{2}}}=\pi } (see also Cauchy distribution)
∫ − ∞ ∞ sin x x d x = π {\displaystyle \int _{-\infty }^{\infty }{\frac {\sin x}{x}}\,dx=\pi } (see Dirichlet integral)
∫ − ∞ ∞ e − x 2 d x = π {\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}} (see Gaussian integral).
∮ d z z = 2 π i {\displaystyle \oint {\frac {dz}{z}}=2\pi i} (when the path of integration winds once counterclockwise around 0. See also Cauchy's integral formula).
∫ 0 ∞ ln ( 1 + 1 x 2 ) d x = π {\displaystyle \int _{0}^{\infty }\ln \left(1+{\frac {1}{x^{2}}}\right)\,dx=\pi }
∫ 0 1 x 4 ( 1 − x ) 4 1 + x 2 d x = 22 7 − π {\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).
∫ 0 1 x 2 ( 1 + x ) 4 1 + x 2 d x = π − 17 15 {\displaystyle \int _{0}^{1}{x^{2}(1+x)^{4} \over 1+x^{2}}\,dx=\pi -{17 \over 15}}
∫ 0 ∞ x α − 1 x + 1 d x = π sin π α , 0 < α < 1 {\displaystyle \int _{0}^{\infty }{\frac {x^{\alpha -1}}{x+1}}\,dx={\frac {\pi }{\sin \pi \alpha }},\quad 0<\alpha <1}
∫ 0 ∞ d x x ( x + a ) ( x + b ) = π agm ( a , b ) {\displaystyle \int _{0}^{\infty }{\frac {dx}{\sqrt {x(x+a)(x+b)}}}={\frac {\pi }{\operatorname {agm} ({\sqrt {a}},{\sqrt {b}})}}} (where agm {\displaystyle \operatorname {agm} } is the arithmetic–geometric mean; see also elliptic integral) Note that with symmetric integrands f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} , formulas of the form ∫ − a a f ( x ) d x {\textstyle \int _{-a}^{a}f(x)\,dx} can also be translated to formulas 2 ∫ 0 a f ( x ) d x {\textstyle 2\int _{0}^{a}f(x)\,dx} .
Efficient infinite series
∑ k = 0 ∞ k ! ( 2 k + 1 ) ! ! = ∑ k = 0 ∞ 2 k k ! 2 ( 2 k + 1 ) ! = π 2 {\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)
∑ k = 0 ∞ k ! 2 k ( 2 k + 1 ) ! ! = 2 π 3 3 {\displaystyle \sum _{k=0}^{\infty }{\frac {k!}{2^{k}(2k+1)!!}}={\frac {2\pi }{3{\sqrt {3}}}}}
∑ k = 0 ∞ k ! ( 2 k ) ! ( 25 k − 3 ) ( 3 k ) ! 2 k = π 2 {\displaystyle \sum _{k=0}^{\infty }{\frac {k!\,(2k)!\,(25k-3)}{(3k)!\,2^{k}}}={\frac {\pi }{2}}}
∑ k = 0 ∞ ( − 1 ) k ( 6 k ) ! ( 13591409 + 545140134 k ) ( 3 k ) ! ( k ! ) 3 640320 3 k = 4270934400 10005 π {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k}}}={\frac {4270934400}{{\sqrt {10005}}\pi }}} (see Chudnovsky algorithm)
∑ k = 0 ∞ ( 4 k ) ! ( 1103 + 26390 k ) ( k ! ) 4 396 4 k = 9801 2 2 π {\displaystyle \sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}={\frac {9801}{2{\sqrt {2}}\pi }}} (see Srinivasa Ramanujan, Ramanujan–Sato series) The following are efficient for calculating arbitrary binary digits of π:
∑ k = 0 ∞ ( − 1 ) k 4 k ( 2 4 k + 1 + 2 4 k + 2 + 1 4 k + 3 ) = 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 }
∑ k = 0 ∞ 1 16 k ( 4 8 k + 1 − 2 8 k + 4 − 1 8 k + 5 − 1 8 k + 6 ) = π {\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)
∑ k = 0 ∞ 1 16 k ( 8 8 k + 2 + 4 8 k + 3 + 4 8 k + 4 − 1 8 k + 7 ) = 2 π {\displaystyle \sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {8}{8k+2}}+{\frac {4}{8k+3}}+{\frac {4}{8k+4}}-{\frac {1}{8k+7}}\right)=2\pi }
∑ k = 0 ∞ ( − 1 ) k 2 10 k ( − 2 5 4 k + 1 − 1 4 k + 3 + 2 8 10 k + 1 − 2 6 10 k + 3 − 2 2 10 k + 5 − 2 2 10 k + 7 + 1 10 k + 9 ) = 2 6 π {\displaystyle \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)=2^{6}\pi }
Plouffe's series for calculating arbitrary decimal digits of π:
∑ k = 1 ∞ k 2 k k ! 2 ( 2 k ) ! = π + 3 {\displaystyle \sum _{k=1}^{\infty }k{\frac {2^{k}k!^{2}}{(2k)!}}=\pi +3}
Other infinite series
ζ ( 2 ) = 1 1 2 + 1 2 2