Spritle, if you are using natural units (radians) then you don't put a unit on angular measures.
For instance, the length "s" of an arc is:
s = theta * r
where theta is the included angle and r is the radius.
Analyze the units
[m] = [1] * [m]
Theta is dimensionless - measured in natural units (radians). So you use a 1.
If we explicitly put radians in there the dimensional analysis would look this:
[m] = [radians] * [m]
Which is an inequality.
And in case you you doubt the equation (s = theta * r ) is true, consider an arc with included angle 2*pi, also known as a circle
s = 2*pi * r = pi*D (where 2*r = D, the diameter)
The result is also known as the circumference of a circle. I'm sure you've seen circumference = pi*D in elementary geometry studies. If not, get something round, measure its diameter, then wrap a string around it, mark the circumference on the string, unwrap it, measure the circumference, and calculate the radio of the circumference to the diameter.