Prism into Pyramids

Euclid determined the volume of a pyramid by comparing it to a prism with the same height and a congruent base. He began by considering a triangular prism and a triangular pyramid.

It's easy to see two pyramids in a prism, and they have congruent bases and the same height (why?), so have equal volume. The Cavalieri Principle says that two solids have equal volume if all corresponding cross sections have equal area. You can check that on the sketch.

Since they don't take up the whole prism, the volume of each must be less than half of the prism. But what shape is the rest of the prism? Try to visualize it before you show it.

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John Golden, Created with GeoGebra