Approximations of the Number π Using Onscribed Regular Polygons
Mieczysław Szyszkowicz *
112 Four Seasons Drive, Ottawa, K2E 7S1, Canada.
*Author to whom correspondence should be addressed.
Abstract
Circa 255 B.C., Archimedes invented a method for approximating the value of the number π. He used the perimeters of the inscribed and circumscribed regular polygons to approximate the perimeter of a circle. Starting with two regular hexagons, he doubled the number of their sides up to 96. This approach allowed him to obtain lower and upper estimations of π. He showed that its value lies in the interval [3 + 10/71, 3 + 1/7]. Here the use of onscribed regular polygons is proposed for a similar purpose. The onscribed regular polygons are placed between the two polygons used in Archimedes’ method. Their location is unique and well defined by applying a criterion to minimize distances. The sequences of areas and perimeters produced by these regular polygons, and their linear combinations, generate values which better approximate π than many other geometrical methods.
Keywords: Archimedes, approximation, circle, polygon, π, quadrature