Random Polygons and Estimations of π

Abstract

Abstract In this paper, we study the approximation of π through the semiperimeter or area of a random n-sided polygon inscribed in a unit circle in ℝ2. We show that, with probability 1, the approximation error goes to 0 as n → ∞, 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.