Proving the length of a circle's arc is proportional to the size of the angle How can I prove that:


  The length of the arc is proportional to the size of the angle.


Every book use this fact in explaining radians and the fundamental arc length equation $s = r\theta$. However no book proofs this fact.

Is this fact some axiom, some natural law like $\pi$ and the triangle side proportions?

Can I proof the above? Or is it something that you should just accept?
I believe this is a matter of definition. (One) Radian measure is defined as the angle subtended at the centre by an arc of length equal to the radius of a circle. So for any arc of length $l$, the angle (in radians) is $\theta = \dfrac l r$. So $ l = r\cdot\theta $
Let o = theta(angle subtended by the arc)
Ur equation can be written as s/r=o

We surely have two radii of circle subtending that angle o and forming arc s.
In the space between them we can fit more such radii.

The number of that radii will be decided by theta and they will form arc s.
Now divide arc s with the radius to get theta.