How Archimedes showed that $\pi$ is approximately equal to 22/7

TL;DR

This paper explores how Archimedes approximated π using the fraction 22/7, detailing his original reasoning and providing an improved lower bound for π.

Contribution

It presents a detailed analysis of Archimedes' method for approximating π and offers an improved lower bound beyond his original bounds.

Findings

Archimedes proved 223/71 < π < 22/7

The paper provides an improved lower bound for π

Historical analysis of Archimedes' approximation method

Abstract

The ratio of the circumference, C, of a circle to its diameter, D, is a constant number denoted by $\pi$ and is independent of the size of the circle. It is known that $\pi$ is an irrational number and therefore cannot be expressed as a common fraction. Its value is approximately equal to 3.141592. Since Archimedes was one of the first persons to suggest a rational approximation of 22/7 for $\pi$, it is sometimes referred to as Archimedes' constant. In this article, we discuss how Archimedes came up with his formula. Archimedes in fact proved that 223/71 < $\pi$ < 22/7. Here we provide an improved lower bound.