Solid angles and particle velocities

[quarky2001]

This should be a simple question, but I haven't found a clear explanation anywhere yet.

Suppose that there are a bunch of particles in a gas, with their velocities "uniformly distributed over solid angles", and I want to find out what fraction of particles are traveling with velocities in a certain angular range.

Conceptually, I know what this means. I'm just not sure how to do it - I drew a diagram.

Suppose the shaded area represents 20% of the sphere's total surface area. How do find the angle theta?

 

[tiny-tim]

hi quarky2001!

(have a theta: θ )

that's called a cap of a sphere of radius θ …

i can't remember the formula for its area , but it's fairly easy to find by integration