$\pi$ and Arc-Length

TL;DR

This paper rigorously derives the existence of π and the concept of arc-length within Euclidean geometry, establishing their fundamental relationships and proving the convergence of arc-length integrals.

Contribution

It provides a geometric proof of π's existence and the derivation of arc-length from Euclidean principles, including the equivalence of circle definitions.

Findings

Proves π's existence using classical geometric definitions.

Shows arc-length is deducible from Euclidean geometry.

Establishes convergence of the arc-length integral.

Abstract

We use the classical definitions (i) $\pi$ is the ratio of area to the square of the radius of a circle; (ii) $\pi$ is the ratio of circumference to the diameter of a circle, to prove $\pi$'s existence within the purview of Euclidean geometry. Next we show that the "arc-length" (Definition 1) is deducible from Euclidean geometry. Then we prove the Non-Euclidean-Axioms(NEA) of Archimedes (Corollary 4 and 5) and that the arc-length integral converges to the arc-length. We justify why `Euclidean Metric' (Definition 5) is a correct metric for arc-length; derive expressions for area, circumference of a circle and finally prove the equivalence of definitions (i) and (ii).