Math
Why is squaring the circle impossible?
Pi is transcendental: it is not the root of any polynomial with rational coefficients. That is why 'squaring the circle' with compass and straightedge is impossible.
Source: Lindemann (1882).
Context
Because pi is transcendental: it is not the root of any polynomial with rational coefficients. Ferdinand von Lindemann proved this in 1882.
Compass and straightedge can only construct lengths that are roots of such polynomials. Squaring the circle needs a length involving the square root of pi, so it cannot be done, however clever the construction.
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