Originally, SI units included radians and steradians as base units (now they're considered derived). The fact that they bothered to include radians and steradians is confusing to me. I usually think of the SI system as trying to be quite minimal. For example, there's no unit for square meters; you just square the meter.
So does the fact that they listed steradians separately from radians imply that a steradian is inherently different from a squared radian?
Like... suppose you decided to measure square angles in terms of the surface area on a sphere swept by an arc of $x$ radians. Initially the area swept by $x$ would be close to $\pi x^2$, but as you approached $x \rightarrow 2\pi$ the swept area would not be increasing very fast anymore and instead of sweeping $\pi x^2 = \pi^3$ you end up sweeping $\pi 4$. Clearly this particular measure of angle area fails to simply be the square of an angle, since it ends up only increasing linearly due to wrapping around the sphere.
Another example. Suppose you defined an angle measure as "area divided by radius". This is a bad measure, since doubling the size of your sphere will double your measure. But it also wouldn't be radians squared.
Are steradians just area over radius squared, or is there something more? When can I confidently multiply measures in radians and say they are in steradians? Are they like meters, where multiplying any two values in meters gives a value in square meters (e.g. a perimeter times a diameter gives you a value measured in square meters, though it may not be the area of the shape)?
If I have an equation that multiplies two values measured in radians, is the result measured in steradians?