Summary: In this paper, we study the approximation of \(\pi\) through the semiperimeter or area of a random \(n\)-sided polygon inscribed in a unit circle in \(\mathbb{R}^2\). We show that, with probability 1, the approximation error goes to 0 as \(n\rightarrow\infty\), and is roughly sextupled when compared with the classical Archimedean approach of using a regular \(n\)-sided polygon. By combining both the semiperimeter and area of these random inscribed polygons, we also construct extrapolation improvements that can significantly speed up the convergence of these approximations.