I was sitting the other day with my cousin, helping him with his homework, when he asked me what's the deal with these radians.

So I took a paper and a pen and tried to recall where did these radians come from, but something is not quite clear for me.

Consider the following circle:

When the angle is 90 degrees, arc

**a** is exactly $\displaystyle \frac{1}{4}$ the length of the full circle's perimeter.

and therfor the angle in radians is:

$\displaystyle \frac{1}{4}\cdot\2\pi r$

So when r=1, we're good, we get: $\displaystyle \frac{1}{4}\cdot\2\pi = \frac{\pi}{2}=90^{\circ}$

But obviously, it is not the case for any other r, as it should be...

So what am I missing here?