The following is a list of significant formulae involving the mathematical constant . Many of these formulae can be found in the article Pi, or the article Approximations of.
where is the circumference of a circle, is the diameter, and is the radius. More generally,
where and are, respectively, the perimeter and the width of any curve of constant width.
where is the area of a circle. More generally,
where is the area enclosed by an ellipse with semi-major axis and semi-minor axis .
where is the circumference of an ellipse with semi-major axis and semi-minor axis and are the arithmetic and geometric iterations of , the arithmetic-geometric mean of and with the initial values and .
where is the area between the witch of Agnesi and its asymptotic line; is the radius of the defining circle.
where is the area of a squircle with minor radius , is the gamma function.
where is the area of an epicycloid with the smaller circle of radius and the larger circle of radius (), assuming the initial point lies on the larger circle.
where is the area of a rose with angular frequency () and amplitude .
where is the perimeter of the lemniscate of Bernoulli with focal distance .
where is the volume of a sphere and is the radius.
where is the surface area of a sphere and is the radius.
where is the hypervolume of a 3-sphere and is the radius.
where is the surface volume of a 3-sphere and is the radius.
Sum of internal angles of a regular convex polygon with sides:
Area of a regular convex polygon with sides and side length :
Inradius of a regular convex polygon with sides and side length :
Circumradius of a regular convex polygon with sides and side length :
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 b<sup>2N</sup>m, when struck by the other object. (This gives the digits of π in base b up to N digits past the radix point.)
Note that with symmetric integrands , formulas of the form can also be translated to formulas .
The following are efficient for calculating arbitrary binary digits of :
Plouffe's series for calculating arbitrary decimal digits of :
In general,
where is the th Euler number.
The last two formulas are special cases of
which generate infinitely many analogous formulas for when
Some formulas relating and harmonic numbers are given here. Further infinite series involving ÃÂ are:
where is the Pochhammer symbol for the rising factorial. See also RamanujanâÂÂSato series.
where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator.
A double infinite product formula involving the ThueâÂÂMorse sequence:
where and is the ThueâÂÂMorse sequence .
One elegant infinite product formula:
where such that .
where is the th Fibonacci number.
For Pythagorean triple (a,b,c).
whenever and , , are positive real numbers (see List of trigonometric identities). A special case is
The following equivalences are true for any complex :
Also
Suppose a lattice is generated by two periods . We define the quasi-periods of this lattice by and where is the Weierstrass zeta function ( and are in fact independent of ). Then the periods and quasi-periods are related by the Legendre identity:
For more on the fourth identity, see Euler's continued fraction formula.
For more iterative algorithms, see the GaussâÂÂLegendre algorithm and Borwein's algorithm.
The symbol means that the ratio of the left-hand side and the right-hand side tends to one as .
The symbol means that the difference between the left-hand side and the right-hand side tends to zero as .
With being the hypergeometric function:
where
and is the sum of two squares function.
Similarly,
where
and 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 and the Fourier coefficients of the J-invariant ():
where in both cases
Furthermore, by expanding the last expression as a power series in
and setting , we obtain a rapidly convergent series for :