Archimedes' Measurement of a Circle

Abstract

Archimedes’ Measurement of a Circle gives one of the earliest scientific approximations to the value of π by describing an iterative process using inscribed and circumscribed polygons within and about the circle based on the angle bisector theorem. Beginning with a regular (inscribed and circumscribed) hexagon, he performs four iterations of angle bisections to generate inscribed and circumscribed 96-gons. In doing so, he also presents a method that could be continued to determine even more precise approximations to π. In this original source lesson we will reconstruct Archimedes’ approach using the angle bisector theorem as the basis for generating the polygons and their corresponding ratios to arrive at his estimates given above. In doing so we will try to identify steps that may never be fully explained as well as acknowledge the fearlessness of Archimedes’ approach.