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.