Pi

Pi is the constant ratio of a circle’s circumference to its diameter, approximately 3.14159. It is irrational and appears throughout geometry, analysis, and probability.

Circle: A plane curve whose points are all equally distant from a fixed center. Pi is defined by comparing a circle’s circumference with its diameter.
Area of a circle: The measure of the region enclosed by a circle, equal to π times the square of its radius. The constant governing circumference also determines circular area.
Rhind Mathematical Papyrus: An ancient Egyptian mathematical document copied by the scribe Ahmes around 1650 BCE. Its circle-area rule implies an approximation to pi near 3.16.
Trigonometry: The mathematical study of relationships between angles and side lengths, and of related periodic functions. Angles in radians and trigonometric periods are expressed using pi.
Tau (mathematical constant): The constant 2π, sometimes used as the angle of one full turn in radians. Tau notation emphasizes a full turn where pi notation uses a half-turn.
Circumference: The length of a closed curve’s boundary, especially a circle’s perimeter. It is the numerator in pi’s defining ratio.
Arc length: The distance measured along a curve between two points. For a circle, arc length equals radius times the angle in radians.
Archimedes' approximation of pi: A method that bounds pi using the perimeters of inscribed and circumscribed regular polygons. Archimedes established rigorous bounds without measuring a physical circle.
Fourier analysis: A method of representing functions as combinations of sinusoidal waves. The standard sine and cosine waves repeat over intervals involving 2π.
Rational approximation: A fraction used to approximate a real number to a chosen degree of accuracy. Fractions such as 22/7 approximate pi but never equal it.

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