Squaring the Circle

Among the many mathematical problems put forward by the ancient Greeks was the problem of squaring the circle - that is, constructing a square with the same area as a given circle, using only a straight edge and compass.
For a circle of radius

\[r\]

and area

\[\pi r^2\]

this means constructing a square of side x such that

\[x^2 = \pi r^2 \rightarrow x= r \sqrt{\pi}\]

.
This was later shown to be impossible as a consequence of the fact that

\[\pi\]

is not the solution of any polynomial equation with rational coefficients (coefficients that can be written as fractions), hence the circle cannot be squared.