I've been staring at a formula for a while now which equates an angle with a solid angle. I can see how the equation is dimensionally consistent, which settles well with me, but I get tripped up at the idea of equating radians with steradians. To me these are two different units.
But apparently it's common for people to mix them all up? In other words, the area of a unit hemisphere is equal to the circumference of a unit circle? See how this doesn't seem to make sense?
Now that I think about it, I suppose a radian ought to be thought of as a ratio, instead of areas and circumferences on unit object. 2$\pi$ radians is the ratio of any sized circle's circumference to its radius. 2$\pi$ steradians is the ratio of any sized hemisphere's area to its radius squared.
Is this the correct way of thinking about it?