Areas of sphere and enclosing cylinder

Suppose you have a sphere in a cylinder which exactly encloses it, that is, the cylinder touches the sphere at the equator and the poles. It turns out that the area of the sphere is the same as the area of the curved part of the cylinder. Earth-shattering this is not, but I'd never come across this before.

I 'discovered' this in the process of trying to solve this puzzle from John Derbyshire's August Dairy at the National Review Online site:

"A cylinder is arranged by the “vertical” tangent lines, which are tangent to the Earth’s sphere at the equatorial points and are parallel to the axis North Pole — South Pole. The surface of the Earth is projected to that cylinder by the rays, which are “emanated” from the said axis and which are parallel to the equatorial plane. Will the area of projection of France be larger or smaller than the area of France itself ?"

I haven't actually solved it yet, but I have a plan ...