Pi
Geometry & MeasurementPi 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 a special number, approximately $3.14159$, that shows up whenever you work with circles. It is the ratio of a circle's circumference (distance around) to its diameter (distance across).
Example
Measure any circle's circumference and divide by its diameter. You always get about $3.14159$. A circle with diameter $1$ cm has circumference exactly $\pi$ cm.
Key Insight
Pi is an irrational number: its decimal goes on forever without repeating. People have calculated trillions of decimal places, but you only need $3.14$ or $22/7$ for most everyday calculations.
Definition
Pi is defined as the ratio $\pi = C/d$ for any circle, where $C$ is the circumference and $d$ is the diameter. Pi is irrational (not expressible as a fraction of integers) and transcendental (not a root of any polynomial with rational coefficients). Its value is approximately $3.14159265358979$.
Example
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 $3.1408 < \pi < 3.1429$.
Key Insight
Pi appears not just in circles but throughout mathematics: in the Gaussian integral ($\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$), in the Basel problem ($\sum 1/n^2 = \pi^2/6$), and in many probability distributions.
Definition
Pi is the unique positive real number satisfying $\sin(\pi) = 0$, or equivalently the half-period of the complex exponential: $e^{i\pi} = -1$ (Euler's identity). Pi is transcendental (proved by Lindemann, 1882), implying the impossibility of squaring the circle with compass and straightedge.
Example
The Leibniz formula: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + \ldots$ (slowly convergent). The Ramanujan series converges much faster: $1/\pi = (2\sqrt{2}/9801) \sum (4k)!(1103+26390k) / ((k!)^4 396^{4k})$.
Key Insight
Euler's identity $e^{i\pi} + 1 = 0$ connects the five most fundamental constants in mathematics. 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.