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

where C is the circumference of a circle, d is the diameter, and r is the radius. More generally, where L and w are, respectively, the perimeter and the width of any curve of constant width. where A is the area of a circle. More generally, where A is the area enclosed by an ellipse with semi-major axis a and semi-minor axis b . where C is the circumference of an ellipse with semi-major axis a and semi-minor axis b and a_n,b_n are the arithmetic and geometric iterations of, the arithmetic-geometric mean of a and b with the initial values a_0=a and b_0=b. where A is the area between the witch of Agnesi and its asymptotic line; r is the radius of the defining circle. where A is the area of a squircle with minor radius r , \Gamma is the gamma function. where A is the area of an epicycloid with the smaller circle of radius r and the larger circle of radius kr , assuming the initial point lies on the larger circle. where A is the area of a rose with angular frequency k and amplitude a . where L is the perimeter of the lemniscate of Bernoulli with focal distance c . where V is the volume of a sphere and r is the radius. where SA is the surface area of a sphere and r is the radius. where H is the hypervolume of a 3-sphere and r is the radius. 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: Area A of a regular convex polygon with n sides and side length s Inradius r of a regular convex polygon with n sides and side length s Circumradius R of a regular convex polygon with n sides and side length s

Physics

A puzzle involving "colliding billiard balls": 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

Note that with symmetric integrands f(-x)=f(x), formulas of the form can also be translated to formulas.

Efficient infinite series

The following are efficient for calculating arbitrary binary digits of π: Plouffe's series for calculating arbitrary decimal digits of π:

Other infinite series

In general, where E_{2k} is the 2kth Euler number. The last two formulas are special cases of which generate infinitely many analogous formulas for \pi when Some formulas relating π and harmonic numbers are given here. Further infinite series involving π are: where (x)_n is the Pochhammer symbol for the rising factorial. See also Ramanujan–Sato series.

Machin-like formulae

Infinite products

where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator. Viète's formula: A double infinite product formula involving the Thue–Morse sequence: where and t_n is the Thue–Morse sequence. Infinite product representation from a limit:

Arctangent formulas

where such that. where F_k is the k-th Fibonacci number. whenever a+b+c=abc and a, b, c are positive real numbers (see List of trigonometric identities). A special case is

Complex functions

The following equivalences are true for any complex z: Also Suppose a lattice \Omega is generated by two periods. We define the quasi-periods of this lattice by and where \zeta is the Weierstrass zeta function (\eta_1 and \eta_2 are in fact independent of z). Then the periods and quasi-periods are related by the Legendre identity:

Continued fractions

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

Iterative algorithms

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

Asymptotics

The symbol \sim means that the ratio of the left-hand side and the right-hand side tends to one as n\to\infty. The symbol \simeq means that the difference between the left-hand side and the right-hand side tends to zero as n\to\infty.

Hypergeometric inversions

With {}_2F_1 being the hypergeometric function: where and r_2 is the sum of two squares function. Similarly, where and \sigma_3 is a divisor function. More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases. Perhaps the most notable hypergeometric inversions are the following two examples, involving the Ramanujan tau function \tau and the Fourier coefficients \mathrm{j} of the J-invariant : where in both cases Furthermore, by expanding the last expression as a power series in and setting z=1/2, we obtain a rapidly convergent series for e^{-2\pi}:

Miscellaneous

Other