TITLE: Why radian is dimensionless?
QUESTION [27 upvotes]: Can't understand why we say that radians are dimensionless. Actually, I understand why this is happening:
theta = arc len / r

Meters/meters are gone and we got this dimensionless. But also we know that angle 57.3 degrees = 1 rad. So, can we use it as dimension?
In such a situation we can say that degrees are also dimensionless, because 1 degree = 1/360 of circle.
How we define the value is dimensionless or not? Why meter is not dimensionless? Where I'm wrong in my conclusions?

REPLY [7 votes]: Yep, you can work with dimensioned angles.
The formula for the arc length is
$$l=\aleph\theta,$$ and that for the area of a sector
$$a=\frac{\aleph\theta r^2}2.$$
The universal constant $\aleph$ is in per-angle-unit, and $\aleph=1 \text{ rad}^{-1}=0.0174533\cdots\text{ deg}^{-1}$.
For example, when considering an harmonic movement,
$$e=A\sin(\aleph\omega t)$$ where $\omega$ is in angle-unit $s^{-1}$ and $e$ in $m$.
$$v=\dot e=A\aleph\omega \cos(\aleph\omega t)$$ is in $ms^{-1}$.

Another universal constant worth to know:
$$\Pi=3.141593\cdots\text{ rad}=180\text{ deg}.$$
denotes the aperture angle of a half-circle. 
It fufills
$$\aleph\Pi=\pi.$$
Hence the famous Euler formula,
$$e^{i\aleph\Pi}=-1.$$