In the diagram above, we have a hemisphere with radius , side by side with a cylinder with radius and height . The cylinder has had an inverted cone with the same radius and height removed from its interior.
The diagram at the right shows the cross-sections of both figures. To change the height of the plane, use the slider at the bottom of the diagram.
Both cross-sections have the same area. Which can be shown using two triangles in the diagram.
Since the cross-sections have the same area at every height of the plane, the two figures have the same volume by Cavalieri's Principle.
Thus,
The volume of the hemisphere is:
Making the volume of a sphere: