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 called , the radius of the base of the cap called , and the height of the cap called , the volume of the spherical cap is then and the curved surface area of the spherical cap is .
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 .
Generally, the volume of a hyperspherical cap of height and radius in -dimensional Euclidean space is given by :
where (the gamma function) is given by .