Triangular Prism Decomposed

This sketch was helpful to students in understanding how the volume of a pyramid relates to a prism.

The prism is shown as its three separate pyramids, regardless of the original shape of the prism.

To show that the pyramids have equal volume, we can use the Cavalieri Principle, which says that solids have equal volume if all corresponding cross sections have equal area. In the sketch we can compare cross sections of the red and the blue, and a different cross section of the green and the blue.

Why are the areas equal?
How can we know that all three volumes are equal if we can only compare pairs?
How does this mean the volume of a pyramid compares to the prism with a congruent base and equal height?

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Extension: Why is it sufficient to show this relationship for triangular pyramids? How does it extend to other polygonal pyramids?

What does this mean about how a cone compares to a cylinder?

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John Golden, Created with GeoGebra