Can Cavalieri's Principle be applied to a Pyramid and a Cylinder?

Question

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.

Answer 1

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.

Answer 2

Cavalieri's principle is an intuitive way of describing Fubini's theorem on multivariate integrals. Archimedes has done marvelous things with it, e.g., a derivation of the volume formula for a sphere using a sphere, a cylinder, and a cone in a unique argument. See the first example here: https://en.wikipedia.org/wiki/Cavalieri%27s_principle