List of formulae involving π
From Wikipedia, the free encyclopedia
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. 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 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, is the gamma function and is the arithmetic–geometric mean.
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:
- Coulomb's law for the electric force in vacuum:
- Approximate period of a simple pendulum with small amplitude:
- Exact period of a simple pendulum with amplitude ( is the arithmetic–geometric mean):
- The buckling formula:
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.[1] (This gives the digits of π in base b up to N digits past the radix point.)