For squaring the circle, it seems easiest to first measure the radius $r$ of the circle, and then trying to construct an arbitrarily good approximation of the length $r \sqrt(\pi)$ using $r$ as the unit length, which must lie in the infinite quadratic extension of the rational. Then use that as the sidelength of the square. An interesting question would be how efficient one could make this process to be. Squaring Circling the circle square can be done using a similar quadratic approximation of $\sqrt(1/\pi)$.
For squaring the circle, it seems easiest to first measure the radius $r$ of the circle, and then trying to construct an arbitrarily good approximation of the length $r \sqrt(\pi)$ using $r$ as the unit length, which must lie in the infinite quadratic extension of the rational. Then use that as the sidelength of the square. An interesting question would be how efficient one could make this process to be. Squaring the circle can be done using a similar quadratic approximation of $\sqrt(1/\pi)$.