Cavalieri's Principle & Pyramids
The volume of a triangular based pyramid is where B = area of the triangular base and H = the height of the pyramid.
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.