Cavalieri's Principle

If two objects can be made from thousands of flat pieces of paper such that each piece of one of the objects has the same area as a corresponding piece of the other object, then the two objects must have the same volume. This is an intuitive way to think of Cavalieri's principle, and it is fundamental to understanding the formula for the volume of a sphere. What we do is compare slices of a sphere with slices of a hollowed-out cylinder. A circular slice from a sphere can have the same area as a ring that is a slice from a cylinder that has been hollowed out by two cones:

More precisely, Cavalieri's Principle says two three-dimensional solids have the same volume if they share the same cross-sectional areas at equal heights.

 

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