That we can't square the circle comes from pi being transcendental. The result that you're thinking of is Galois' proof that there is no algebraic formula forroots of 5th degree polynomials.
Yeah, and constructability is usually handled by proving that a length is constructable if it lives in an iterated quadratic extension of the rationals. Pi does not lie in such an extension, so is not a constructable length (and neither is its square root).
