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Archimedes Volume of a Sphere

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: