Why radian is dimensionless? 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?

A dimensionless quality is a measure without a physical dimension; a "pure" number without physical units.
However, such qualities may be measured in terms of "dimensionless units", which are usually defined as a ratio of physical constants, or properties, such that the dimensions cancel out.  Thus the radian measure of angle as the ratio of arc length to radius length is one where the units of length cancel out.
