Circles and PiSphere Sum

If you compare the equations for the volume of a cylinder, cone and sphere, you might notice one of the most satisfying relationships in geometry. Imagine we have a cylinder with the same height as the diameter of its base. We can now fit both a cone and a sphere perfectly in its inside:

This cone has radius r and height 2r. Its volume is

This sphere has radius r. Its volume is

This cylinder has radius r and height 2r. Its volume is

Notice how, if we ???

the volume of the cone and the sphere, we get exactly the volume of the cylinder!

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