The ratio of the arc lenght to the radius

Question

Using elementary geometry how is proved that the ratio of the arc length to the radius is the same for concentric arcs of different diameters.

Answer 1

It is proved by constructing a similarity of the Euclidean plane, which fixes the center point $O$ of those concentric arcs, and which stretches each ray based at that point by a common factor equal to the quotient $q = \frac{r_2}{r_1}$. One then uses elementary geometry to conclude that the length of every line segment is stretched by that same common factor $q$. By approximating a circular arc with a concatenation of chords, it follows that the length of a circular arc is also stretched by that same factor $q$.

If you were modelling your Euclidean plane with Cartesian coordinates, and if $O$ were the origin of the coordinate system, then this similarity map would simply be $$f(x,y) = (qx,qy) $$

Answer 2

This is a matter of proportions.

$$\frac{\text{arclength}}{\text{circumference}}=\frac{\text{angle}}{2\pi}\implies \frac{s}{2\pi r}=\frac{\theta}{2\pi}\implies \frac{s}{r}=\theta$$