Cavalieri's Principle with Different Pyramids
Cavalieri's Principle states that if two or more objects have the same height and parallel cross sections at any level have the same area, then the objects have the same volume. One application of this is to extend the formula for the volume of a pyramid from triangular based pyramids (which can be derived directly by dissection), to pyramids with any shape base. The volume of a triangular based pyramid is
V = 1/3 A(base)*height.
Since a pyramid of any base shape can be paired with a triangular base pyramid, and since it can be shown that the parallel cross sections at any height have equal area, the volume formula applies regardless of the shape of the base.
1. Move the apex points laterally. Note that the areas of the base and cross sections are not affected. Therefore moving the apex parallel to the base does not affect the volume of a pyramid.
2. Vary the height.
3. Vary the height of the cross sections. Note that the cross sectional areas remain equal.
4. Vary the shape of the base of the third figure. (The construction automatically scales the copy used in the base of the pyramid to have the same area as the triangle and the square. Note that the shape does not affect the scaling of the cross sections, so the pyramid volume formula really does apply to any base shape.