Half of a sphere cut by a plane passing through its center. A hemisphere of radius can be given by the usual spherical coordinates
(1)
| |||
(2)
| |||
(3)
|
where and . All cross sections passing through the z-axis are semicircles.
The volume of the hemisphere is
(4)
| |||
(5)
| |||
(6)
|
The weighted mean of over the hemisphere is
(7)
|
The geometric centroid is then given by
(8)
|
(Beyer 1987).