A zone of a sphere is the part between two parallel planes. If the sphere has radius r and the distance between the planes is h, the area of the curved surface of the zone equals 2πrh.
A circumscribing cylinder of a sphere is a right-circular cylinder with the same radius as the sphere and with its axis passing through the centre of the sphere. Suppose that a zone is formed by two parallel planes. Take the circumscribing cylinder with its axis perpendicular to the two planes as shown. It is an interesting fact that the area of the curved surface of the zone is equal to the area of the part of the circumscribing cylinder between the same two planes. The special case when one of the planes touches the sphere gives the area of a spherical cap as 2πrh.