In this lesson, students use informal arguments to show that the volume of any pyramid is one-third the volume of the prism with equal height and congruent base.

First, students determine that any triangular prism can be split into 3 pyramids of equal volume, and that any triangular pyramid can be manipulated to build such a prism. Therefore, the expression gives the volume of any triangular pyramid.

Next, students consider a cone or pyramid of any particular size, and compare its volume to a triangular pyramid with the same height and a base of equal area. They use cross-section and dilation arguments from earlier lessons to show that Cavalieri’s Principle applies to the two solids, and therefore the volumes are equal. Because any pyramid or cone can be shown to have the same volume as a triangular pyramid with the same height and a base with equal area, the expression extends to all pyramids and cones.

As students compare pairs of pyramids, draw conclusions about their volumes, and extend the ideas to include all pyramids, they are making sense of a problem (MP1).