Zone

The surface area of a spherical segment. Call the radius of the sphere , the upper and lower radii and , respectively, and the height of the spherical segment . The zone is a surface of revolution about the z-axis, so the surface area is given by

(1)

In the -plane, the equation of the zone is simply that of a circle,

(2)

so

			(3)
			(4)

and

			(5)
			(6)
			(7)
			(8)

This result is somewhat surprising since it depends only on the height of the zone, not its vertical position with respect to the sphere.