Why is 1 steradian equal to 1 radian squared?

[JamesGoh]

Im aware that 1 steradian = 1 radian$$^{2}$$ for a solid angle measurement.

Is the reason why 1 steradian = 1 radian$$^{2}$$ because of the 3-D nature of a sphere ?

 

[jambaugh]

Actually the sphere (2 dim) is the surface of the ball (3-dim) but yes.
You can define 1 steradian as the solid angle forming one unit of area on the unit sphere,
just like 1 radian is the angle forming one unit length on the unit circle.