Pi
Pi is the mathematical constant equal to the ratio of any circle's circumference to its diameter, approximately 3.14159.
Formula
\pi = \frac{C}{d} = 3.14159\ldots
Definition
Pi (written as the Greek letter $\pi$) is the ratio of any circle's circumference to its diameter, $\pi = C/d$, approximately $3.14159265358979$. Pi is irrational (its decimal never repeats and cannot be written as a fraction of integers) and transcendental (not a root of any polynomial with rational coefficients, as Lindemann proved in 1882), which implies the impossibility of squaring the circle with compass and straightedge. Equivalently, $\pi$ is the unique positive real number satisfying $\sin(\pi) = 0$, or the half-period of the complex exponential in Euler's identity $e^{i\pi} = -1$.
Example
Measure any circle's circumference and divide by its diameter, and you always get about $3.14159$; a circle with diameter $1$ cm has circumference exactly $\pi$ cm. Archimedes approximated $\pi$ by inscribing and circumscribing regular polygons around a circle; with a $96$-sided polygon he showed $3\frac{10}{71} < \pi < 3\frac{1}{7}$, giving $3.1408 < \pi < 3.1429$. The Leibniz formula $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + \ldots$ converges slowly, while the Ramanujan series $1/\pi = (2\sqrt{2}/9801) \sum (4k)!(1103+26390k) / ((k!)^4 396^{4k})$ converges far faster.
Key Insight
People have calculated trillions of decimal places of $\pi$, but $3.14$ or $22/7$ is enough for most everyday calculations. Pi appears throughout mathematics well beyond circles, in the Gaussian integral ($\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$), the Basel problem ($\sum 1/n^2 = \pi^2/6$), and many probability distributions. Euler's identity $e^{i\pi} + 1 = 0$ connects the five most fundamental constants in mathematics, and the transcendence of $\pi$ means the ratio of a circle's area to a square's area cannot be a simple algebraic number, a profound limitation on geometric constructions.