Editing Squaring the circle (section)

{{Short description|Problem of constructing equal-area shapes}}
{{other uses|Squaring the circle (disambiguation)|Square the Circle (disambiguation)|Squared circle (disambiguation)}}
{{distinguish|Square peg in a round hole}}
{{good article}}
{{Use dmy dates|date=January 2020}}
[[File:Squaring the Circle J.svg|thumb|Squaring the circle: the areas of this square and this circle are both equal to {{pi}}. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized [[compass and straightedge]].]]
{{Pi box}}
'''Squaring the circle''' is a problem in [[geometry]] first proposed in [[Greek mathematics]]. It is the challenge of constructing a [[square (geometry)|square]] with the [[area of a circle|area of a given circle]] by using only a finite number of steps with a [[compass and straightedge]]. The difficulty of the problem raised the question of whether specified [[axiom]]s of [[Euclidean geometry]] concerning the existence of [[Line (geometry)|lines]] and [[circle]]s implied the existence of such a square.

In 1882, the task was proven to be impossible, as a consequence of the [[Lindemann–Weierstrass theorem]], which proves that [[pi]] (<math>\pi</math>) is a [[transcendental number]].
That is, <math>\pi</math> is not the [[zero of a function|root]] of any [[polynomial]] with [[Rational number|rational]] coefficients. It had been known for decades that the construction would be impossible if <math>\pi</math> were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found.

Despite the proof that it is impossible, attempts to square the circle have been common in [[pseudomathematics]] (i.e. the work of mathematical cranks). The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.{{r|idioms}} 

The term ''quadrature of the circle'' is sometimes used as a synonym for squaring the circle. It may also refer to approximate or [[Numerical integration|numerical methods]] for finding the [[area of a circle]]. In general, [[quadrature (geometry)|quadrature]] or squaring may also be applied to other plane figures.