π (sometimes written pi) is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.14159 in the usual decimal notation (see the table for its representation in some other bases). π is one of the most important mathematical and physical constants: many formulae from mathematics, science, and engineering involve π.
π is an irrational number, which means that its value cannot be expressed exactly as a fraction m/n, where m and n are integers. Consequently, its decimal representation never ends or repeats. It is also a transcendental number, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can be equal to its value; proving this was a late achievement in mathematical history and a significant result of 19th century German mathematics. Throughout the history of mathematics, there has been much effort to determine π more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture.
The Greek letter π, often spelled out pi in text, was adopted for the number from the Greek word for perimeter, first by William Jones in 1707, and popularized by Leonhard Euler in 1737
Area of the circle = π × area of the shaded square
In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter:
The ratio C/d is constant, regardless of a circle's size. For example, if a circle has twice the diameter d of another circle it will also have twice the circumference C, preserving the ratio C/d.
Alternatively π can be also defined as the ratio of a circle's area (A) to the area of a square whose side is equal to the radius:
See also: Chronology of computation of π and Numerical...