Squaring the circle

Squaring the circle is the impossible straightedge-and-compass construction of a square with the same area as a given circle. Its impossibility follows from the transcendence of π.

Straightedge and compass construction: A geometric construction using only an unmarked straightedge and compass. These are the only tools allowed in the classical challenge.
Ancient Greek mathematics: The mathematical tradition developed in the Greek-speaking world from antiquity onward. The classical formulation arose within Greek geometry’s construction problems.
Doubling the cube: The classical construction problem of making a cube with twice a given cube’s volume using straightedge and compass. It is another famous impossible Greek construction, blocked by a different algebraic obstacle.
Classical construction problems: Three ancient Greek challenges: doubling the cube, trisecting an angle, and squaring the circle. Its impossibility became part of the resolution of the classical trio.
Constructible number: A real number obtainable from rational numbers by finitely many arithmetic operations and square roots. Straightedge-and-compass lengths correspond to constructible numbers.
Hippocrates of Chios: A Greek mathematician of the fifth century BCE known for work on geometry and quadrature. His quadratures of certain lunes showed that some curved regions could be squared.
Trisecting the angle: The classical problem of dividing an arbitrary angle into three equal parts using straightedge and compass. Like circle squaring, general angle trisection fails under the classical tool restrictions.
Lindemann's proof of the transcendence of π: Ferdinand von Lindemann’s 1882 proof that π is transcendental. The proof converted a geometric puzzle into a consequence of number theory.
Transcendental number: A number that is not a root of any nonzero polynomial with integer coefficients. A transcendental π cannot be a constructible length.
Lunes of Hippocrates: Two-moon-shaped regions bounded by circular arcs whose combined area can equal a polygon’s area. These successful quadratures encouraged investigation of whether a circle itself could be squared.

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