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 nonalgebraic, 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".
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