During my studying of physics, I've been introduced to a concept of a solid angle. I think that I do understand it pretty good, however, I'm stuck with one certain problem.
We know that a solid angle is $S/r^2$ where $S$ is the area subtended by a cone with the vertex in the center of a sphere with radius $r$.
Suppose we have some arbitrary surface that encloses some volume. And suppose I want to insert a cone into that surface, such that the cone will cover/cut some tiny area $\Delta S$ on that surface (that will be its base). I want to know the solid angle that subtends this surface.
I do not understand, why I should take the projection of $\Delta A_2$ here (which is a vector for area $\Delta S_2$), in order to calculate the solid angle $\Delta \Omega$. Why they claim $\Delta A_2 \cos \theta$ is "the radial projection of $\Delta A_2$ onto a sphere $S_2$ or radius $r_2$"? What if the area is not necessarily sphere-like? Is there any mathematical proof of this?