ALL measures of angles, not just radians, are dimensionless. They come
essentially from ratios of lengths, like the sine or tangent.
The radian is just the ratio of arc length (in say meters) to radius (in
meters). This ratio stays the same whether one measures in meters, or
centimeters or feet. Thus the angle is dimensionless. The number you get
is independent of what system of units you use.
J. Epstein