Defining angle and showing that the measure of an angle is well defined. An angle is the figure formed by two rays with common endpoint. For each pair of line segments that share a common endpoint we associate an angle by extending each segment away from their common endpoint so that both segments have rays extending from them. Then, the angle formed by the rays is what we call the angle formed by the segments.
To assign a measure to an angle: From the common endpoint, which we refer to as a vertex, we draw a circle of radius $r$ centered at the vertex. Let the part of the circle (How to rigorously define which arc you want to look at?? have length $s$.
Then the measure of the angle (in the unit of radians) is defined as,
$$\theta=\frac{s}{r}$$
How can we show that the measure of the angle doesn't depend on the radius of the circle chosen?
I guess I can do it with calculus, look at arc lengths etc. But is their a more elementary way?
 A: The definition of an angle I know is the definition for an angle between two vectors, via
$$
\theta = \arccos\left(\frac{\langle u, v\rangle}{\|u\|\|v\|}\right).
$$
In this case, the vectors are normalized, so that's why it doesn't depend on the radius. 
I hope that's satisfactory, despite it not being very geometric. 
A: In euclidean geometry we have the notion of similarity or scaling. Under a scaling all distances or lengths of segments are multiplied by the same factor $\lambda>0$. Since the length of a smooth curve is the sup of the length of inscribed polygons the lengths of curves are multiplied by the same factor $\lambda$. This allows to conclude that the measure of an angle as defined in the question does not depend on the radius $r$ of the circle drawn.
A: Because the length of the arc of a given angle DEPENDS on the radius of the circle. If the radius doubles, the arc length also doubles. Say you have an angle $\theta = \frac{pi}{2}$ in radians ( = 90°) measured on a circle with  radius r1 . 
The arc length equals $\frac{2\pi r1}{4} = \frac{\pi}{2} * r1$ (Def of 1 radian: The magnitude of the angle that measures an arc length of r on the circle)
so $$\frac{\pi}{2} rad =\frac{\frac{\pi}{2}∗r1}{r1} = \frac{\pi}{2}  $$ So it just cancels out.
if now you draw a circle around the first circle with $r2 = 2 \space r1$
$$\frac{\pi}{2} rad =\frac{\frac{\pi}{2}∗r2}{r2} =\frac{\frac{\pi}{2}∗2r1}{2r1} = \frac{\frac{\pi}{2}∗r1}{r1} = \frac{\pi}{2}$$
Both the radius AND the Arc length double so how many times the radius 'fits' in one arc length - which is the magnitude of the angle - doesn't change.
