In a previous lesson, students made a conjecture about the relationship between the volume of a triangular pyramid and the volume of a prism with the same height and a base congruent to the pyramid’s.
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 \(\frac13 Bh\) 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 \(\frac13 Bh\) 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).
- Use decomposition and Cavalieri’s Principle to informally justify (using words and other representations) the volume formula $V=\frac13Bh$ for pyramids.
- Let’s create a formula for the volume of any pyramid or cone.
If not already done in the previous lesson, consider creating an extra few sets of nets and pyramids ahead of time. This ensures that all groups will have pyramids to work with even if the pyramids were accidentally discarded between lessons.
Be prepared to display an applet for all to see in the activity Splitting a Prism into Pyramids.
- I can explain why the volume formula for pyramids and cones is $V=\frac13Bh$.
The single point on a cone or pyramid that is the furthest from the base. For a pyramid, the apex is where all the triangular faces meet.