I've heard of the ancient problem of squaring the circle, whereas a circle is made into a square. I've heard that since $\pi$ is NOT rational (and a transcendental) number, it is not construct-able. my question is, is there a close estimate that will be close to $\pi$? as in:
$$ f(x)\sim \pi$$
Yes, essentially you'd want to take the partial continued fraction expansions of $\pi$ and continually construct squares of that side length. There are lots of ways to use continued fractions to do this, my favorite is the following;
$$ \pi = \cfrac{4}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\cfrac{4^2}{9+\ddots}}}}} $$