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. where C is the circumference of a circle, d is the diameter, and r is the radius.
are the arithmetic and geometric iterations of
, the arithmetic-geometric mean of a and b with the initial values
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,
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 SV is the surface volume of a 3-sphere and r is the radius.
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: 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.
Note that with symmetric integrands
The following are efficient for calculating arbitrary binary digits of π: Plouffe's series for calculating arbitrary decimal digits of π:[6] In general, where
[9] The last two formulas are special cases of which generate infinitely many analogous formulas for
Some formulas relating π and harmonic numbers are given here.
Further infinite series involving π are:[15] where
is the Pochhammer symbol for the rising factorial.
Viète's formula: A double infinite product formula involving the Thue–Morse sequence: where
are positive real numbers (see List of trigonometric identities).
A special case is The following equivalences are true for any complex
We define the quasi-periods of this lattice by
is the Weierstrass zeta function (
Then the periods and quasi-periods are related by the Legendre identity: For more on the fourth identity, see Euler's continued fraction formula.
is the sum of two squares 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
of the J-invariant (OEIS: A000521): where in both cases Furthermore, by expanding the last expression as a power series in and setting
, we obtain a rapidly convergent series for