Showing the projection from the sphere to the cylinder is area preserving

Question

Let $S$ be the complement of the points $(0,0,\pm{1})$ in $S^2$, and let $C=\{(x,y,z) \mid x^2+y^2 = 1,\ |z|< 1 \}$, be a cylinder of radius $1$. If $\varphi : S\to C$ is the map given by radial projection from the $z$-axis, show that $\varphi$ is area-preserving.

Answer 1

By rotational symmetry, it is enough to show that the area of an annulus (points with latitude betwee $\alpha$ and $\beta$) is porportional to its height $\sin\beta-\sin\alpha$. If $\alpha\approx\beta$, this can be approximated by the lateral surface of a chopped off cone, which again is proportional to the circumference $\approx 2\pi \cos\alpha$ times the tilted height $\beta-\alpha$. Ultimately this boils down to the fact that the derivative of $\sin$ is $\cos$ (or the integral of $\cos$ is $\sin$, depending on perspective)