## Squaring the Circle

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.