Modernizing Archimedes' Construction of $\pi$

TL;DR

This paper revisits Archimedes' classical method for measuring π, providing a rigorous modern geometric framework that integrates differential properties of trigonometric functions into introductory calculus.

Contribution

It modernizes Archimedes' construction of π by formalizing his measurement procedure with rigorous geometric and differential analysis.

Findings

Provides a rigorous geometric interpretation of Archimedes' method

Integrates differential properties of trigonometric functions into calculus

Enhances understanding of π's measurement in an educational context

Abstract

In his famous work, "Measurement of a Circle," Archimedes described a procedure for measuring both the circumference of a circle and the area it bounds. Implicit in his work is the idea that his procedure defines these quantities. Modern approaches for defining $\pi$ eschew his method and instead use arguments that are easier to justify, but they involve ideas that are not elementary. This paper makes Archimedes' measurement procedure rigorous from a modern perspective. In so doing, it brings a rigorous and geometric treatment of the differential properties of the trigonometric functions into the purview of an introductory calculus course.