The fact that surface of spherical segment only depends on its height follows from symplectic geometry It is quite quite well known that the surface of the piece of a sphere with $z_0<z<z_1$ for some values of $z_0,z_1$ is given by 
$ S = 2\pi R (z_1-z_0) $. So this surface area only depends on the height of the spherical segment. 
My teacher mentioned that this fact follows from some general theorem in symplectic geometry. He mentioned this right after giving the action of $U(1)$ on $S^2$ by rotation as an example of a hamiltonian Lie group action. So I guess that it should follow somehow from this action. (Whose moment map is essentially the z-coordinate of the sphere.)
What could be the general theorem my teacher was talking about?

It follows from the Duistermaat-Heckman theorem as described here 
