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.
Highlights
The classical approach to estimate π, the ratio of the circumference of a circle to its diameter, based on the semiperimeter of regular polygons inscribed in or circumscribed about a unit circle in R can be traced to Archimedes more than 2000 years ago [1]
In this paper, we study the approximation of π through the semiperimeter or area of a random nsided polygon inscribed in a unit circle in R
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
Summary
The classical approach to estimate π, the ratio of the circumference of a circle to its diameter, based on the semiperimeter (or area) of regular polygons inscribed in or circumscribed about a unit circle in R can be traced to Archimedes more than 2000 years ago [1]. Note that such random polygons will rarely be regular (when the vertices happen to be all spaced on the circle), it is intuitively clear that, as n becomes large, these random vertices tend to spread out and become “evenly” distributed on the circle so that the semiperimeter or area of the circle may still be well approximated by the corresponding semiperimeter or area of the inscribed random polygon. This is con rmed by the strong convergence results stated in the theorem below. We will show that, for both Archimedean and our random approximations of π, by applying extrapolation type techniques [3], it is possible to construct some simple linear combinations of Sn and An that can greatly improve the accuracy of these approximations
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