Question

Hi there! In the reading notes "Units and Dimensional Analysis" (8.01SC - Unit 1), about steradians, it's shown that the solid angle for a sphere is 4*pi. It's said then that this result doesn't depend upon the position of the angle's vertex, as long as the vertex is inside the sphere.
Doesn't the steradians definition imply that the vertex is located on the sphere's centre?

Answer 1

Definition of steradians is not the issue here. Here you conclude that what applies at sphere's center applies for any point inside of the sphere. (very powerful thing)

If you think in 2D, draw a circle. Any point inside of a circle is encircled by the circle i.e. angle is 2pi. Same thing for 3D and a sphere.

Note however, if you have a semicircle in 2D, then the angle does depend on position of a vertex and it is not always 1pi but varies from 0 (vertex at infinity) to 1pi (vertex in the center of the semicircle. Same thing by analogy applies to your question in 3D. And therefore you cannot derive the same conclusion for the semicircle.

Answer 2

I think I got the idea now. Thank you very much.