How to prove $C=2πr$?

Question

Everybody know this formula,but why the relation between $C$ and $r$ is linear relation? Not $C=2πr^{0.99}$ or $C=2πr^{1.01}$how to prove it,what axiom is it based on?

Answer 1

As the rectification of the circle ultimately relies on triangulation, this follows from the corresponding proportionality for triangles, hence ultimately from the intercept theorem (not a single axiom).