I would like to ask a basic question related to circular arcs or angles.
It is a well known fact that the ratio of circumference to the radius, in any circle, is constant (and is equal to $2\pi$).
I suppose that this fact makes us to define the concept of angles or angular distance. Referring the given figure. Let the inner circle be a unit circle. Now we assign some number to the rotation from $OA$ to $OB$, usually denoted by $\angle BOA$.
We define $\angle BOA$ as $$ \angle BOA = \text{length}\ \overset{\huge\frown}{AB}. $$
But this definition would make sense if (let's suppose the radius of the outer circle is $2$) $$\frac{\text{length}\ \overset{\huge\frown}{CD}}2=\text{length}\ \overset{\huge\frown}{AB}.$$ as the rotation from $A$ to $B$ is same as from $C$ to $D$.
This is precisely my question: how do we know that $\text{length}\ \overset{\huge\frown}{CD}=\text{length}\ \overset{\huge\frown}{AB}$?
In other words, we know that for both the circles, the ratio of circumference to the radius is same but how do we know that the ratio of arc lengths $\overset{\huge\frown}{AB}$ and $\overset{\huge\frown}{CD}$ to their respective radii is same?