Cutting a sphere
A sphere, a cone and a cylinder of same radius are cut by the same horizontal plane. The height of the cone and cylinder is equal to their radius.
We can compare their cross sections obtain from the height.
Drag G to change the height of the cutting plane.
Regardless of the height of the cutting plane, the sum of sections from the sphere and cone is the equal to the section of the cylinder.
By the Cavalieri's Principle*, the volume of the cylinder should be equal to the sum of volume of the cone and the hemisphere.
* http://en.wikipedia.org/wiki/Cavalieri's_principle