# Area Inside A Circle: Intuitive and Rigorous Proofs

## TL;DR

This paper reviews various proofs of the circle's area, highlighting pedagogical issues and introducing an innovative theorem to establish the area without circular reasoning.

## Contribution

It presents a new approach with a theorem that avoids circular reasoning in proving the area of a circle, improving mathematical pedagogy.

## Key findings

- Traditional proofs often use circular reasoning.
- The new theorem provides a non-circular proof method.
- Enhanced clarity in teaching circle area proofs.

## Abstract

In this article I conduct a short review of the proofs of the area inside a circle. These include intuitive as well as rigorous analytic proofs. This discussion is important not just from mathematical view point but also because pedagogically the calculus books still today use circular reasoning to prove the area inside a circle (also that of an ellipse) on this important historical topic, first illustrated by Archimedes. I offer an innovative approach, through introduction of a theorem, which will lead to proving the area inside a circle avoiding circular argumentation.