Spherical Cap

In geometry, a spherical cap is a portion of a sphere cut off by a plane. If the plane passes through the center of the sphere, so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere.

If the radius of the sphere is, the radius of the base of the cap is, and the height of the cap is, then the volume of the spherical cap is

and the curved surface area of the spherical cap is

The relationship between and are irrelevant as long as and . The blue section of the illustration is also a spherical cap.

The parameters, and are not independent:

.

Substituting this into the area formula gives:

Note also that in the upper hemisphere of the diagram, and in the lower hemisphere ; hence in either hemisphere and so an alternative expression for the volume is

.