Squaring the circle

The quadrature of the circle, better known as squaring the circle, is a classical problem of mathematics, or more specifically, of geometry. The problem is to construct, using only ruler-and-compass constructions, a square with the same area as a given circle.

The square and circle have the same area

This problem dates back to the invention of geometry, but only during the late 1800s it was possible to show that this construction was impossible. A solution demands construction of the number <math>\sqrt{\pi}</math>, and the impossibility of this undertaking follows from the fact that <math>\pi</math> is a transcendental number, i.e. it is non-algebraic, and only algebraic numbers may be constructed with ruler and compasses alone.

If you solve the problem of the quadrature of the circle, this means you have also found an algebraic value of π -- this is impossible.

"Squaring the circle" as a metaphor

The mathematical proof that the quadrature of the circle is impossible has not hindered many "free spirits" to invest years in this problem anyway.

The futility of undertaking exercises aimed at finding the quadrature of the circle has brought this term into use in totally unrelated contexts, where it is simply used to mean "a hopeless/meaningless/vain undertaking".