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?
Best Answer
The radian measures an angle in the plane (2 dimensional space). The steradian measures a vertex in 3 dimensional space. If you had a pyramid, for example, you could use radians to measure each of the angles at the vertex, and steradians to measure the vertex as a whole.
To measure a vertex in steradians, you would imagine a unit sphere with the vertex at the center, and the measure the area of the sphere inside the vertex.
There is an alternative way to measure it. If the intersection of the vertex and the sphere forms a polygon. If you measure the angels and sum them, and compare that to what that number would be for a polygon with the same number of sides on a plane, the difference is the measure of the vertex.
If you had a regular tetrahedron. The measure of each angle at the vertex is $\frac \pi3$ radians, but the vertex is $3\cos^{-1} \frac 13 - \pi$ steradians.