## Pyramid Volume in SketchUp ☆

Frequently, students’ first instinct about the volume of a pyramid is that is must be 1/2 the volume of a prism. While there are many ways of helping students understand the correct relationship, including conducting an experiment to see how many times “filling” a pyramid, say with rice, is needed to fill a prism (with the same base shape and height), the following uses SketchUp (a free 3-D modeling software program) to convey this idea.

(password: mathematicalmusings)

Technically, since the most common objects of study are *right* pyramids, the argument from this visualization, which depicts three oblique pyramids, relies on Cavalieri’s principle (which concludes that the volume of right and oblique pyramid – same base shape and height – have equal volumes by comparing the areas of every cross section). In addition, while for a cube the three pyramids will be identical, for the right rectangular prism depicted in the video, the three pyramids are all different, yet all have the same volume (based on the three lengths/dimensions being the same). This means that the volume of a right pyramid is one-third of its prism.

While this visualization works nicely for rectangular pyramids, the exact same visual does not necessarily help with other polygonal (e.g., hexagonal) pyramids or cones, since extracting three identical pyramids from one of those prisms is more complicated. Yet for helping students understand the overarching relationship between the volumes of right prisms and pyramids, the visualization of the rectangular prism and pyramids does provide some meaningful validation of the relationship.