I know that Cavalieri's Principle makes it so that if two prisms/cylinders, or two pyramids/cones have the same area at a cross section parallel to the base, and they have the same height, they also have the same volume. However, does it still apply to a pyramid/cone and a prism/cylinder?
Research:
All examples of Cavalieri's Principle show two prisms/cylinders or two pyramids/cones. They never get mixed. I have found that on Wolfram Mathworld it is defined as "…the same distance from their respective bases are always equal…" implying that a cone and a prism wouldn't fall under this since the prism's cross-section would remain constant, but the cone's would increase or decrease based on height.
Best Answer
As you point out, if we're comparing a solid with constant cross-sectional area to one whose cross sectional area vanishes as we move away from its base, then Cavalieri's Principle is inapplicable. Put another way, a pyramid/cone might have the same volume and height as a prism/cylinder, but even in such a case, they cannot have equal cross-sectional areas throughout.