Archimedes' Hat-Box Theorem

Enclose a sphere in a cylinder and cut out a spherical segment by slicing twice perpendicularly to the cylinder's axis. Then the lateral surface area of the spherical segment is equal to the lateral surface area cut out of the cylinder by the same slicing planes, i.e.,

where is the radius of the cylinder (and tangent sphere) and is the height of the cylindrical (and spherical) segment.