In particular for the squaring, it is easy to calculate the relation between the radius $r$ of the circke and the side $l$ of the square with the same area of the starting circle: \[L = \sqrt {\pi} r \] Now, since $\pi$ is a transcendental number, the formula above is the simplest representation of the impossibility of squaring the circle using ruler and compass: with these devices you can treat rational and irrational numbers, such as $\sqrt{2}$ (in this case simply draw a square of side 1).
So, using these two tools it is possible to obtain an approximate construction and, therefore, a corresponding approximate value for $\pi$: during the XX century there are produced a lot of approximations, for example by CD Olds (1963), Martin Gardner (1966), Benjamin Bold (1982). They are all variations of the geometric construction discovered by Srinivasa Ramanujan in 1913, that approached $\pi$ with the fraction \[\frac{355}{113} = 3.1415929203539823008 \dots\] right up to the sixth decimal place.
In 1914, Ramanujan discovered a more accurate approximation (to eight decimal places), always using ruler and compass: \[\left (9^2 + \frac{19}{22}^2 \right)^{1/4} = \sqrt[4]{\frac{2143}{22}} = 3.1415926525826461252 \dots\]
