Calculating the Solid Angle Subtended by a Disc: How to Find ω?

[brotherbobby]

The problem is to find out the solid angle ω subtended by a disc of radius a at a point P distant z from its centre along its axis. α is the semi-vertical angle of the disc at the point P in question.

The answer is supposed to be ω = 2π (1 - cos α), according to an online text. However, I find that ω = (Area) / (perpendicular distance)² = (πa²)/z² = π tan²α.

I mark my answer in red, in contrast to the "correct" answer in blue.

Any help?

 

[soarce]

"Area" is not the area of the disc, but the area delimited by the disk on the surface of the sphere with the center in P.

 

[brotherbobby]

Thanks mate.

Now how do I find this area?

 

[soarce]

I think it is a good exercise to calculate it: