Volume of a Pyramid

In grade 9 we are supposed to develop the formula for the volume of a pyramid via investigation. That is, by doing some sort of procedure (not algebraically). If you are lucky your schools might have a set (or several sets) of these volumetric solids so you can pour liquids from one to another to compare but if not, here is a cheap way to do the same thing (at least for right prisms). For an idea of how to do it via water and the volumetric solids, check out this post from Tap Into Teen Minds.

• MPM1D, MFM1P - develop, through investigation (e.g., using concrete materials), the formulas for the volume of a pyramid, a cone, and a sphere (e.g., use three-dimensional figures to show that the volume of a pyramid [or cone] is 1/3 the volume of a prism [or cylinder] with the same base and height, and therefore that V_pyramid= V_prism/3

• Each student or group needs six of the large nets to make their rectangular prism and three of the small nets to make the cube. See below to download the nets.
• Note that each of these shapes is a pyramid, just non symmetrical.

Note that these instructions are for students. You may just wish to do this as a demo.

1. Have the students construct the nine shapes using the nets (6 large and 3 small). 
2. Taking the six large shapes, construct three symmetrical pyramids. Note the dimensions (size of base and height).
3. Using the same six shapes, create a rectangular prism. Note the dimensions (size of base and height).
4. How do the height and base of the prism compare to the one of the symmetrical pyramids? 
5. How do the volumes of each pyramid compare to that of the prism?
6. Taking the three small shapes, note the type of solid and its dimensions (size of base and height)
7. Using the same three shapes construct a cube. Note the dimensions (size of base and height)
8. How do the height and base of the cube compare to each non symmetrical pyramid? 
9. How do the volumes of each non symmetrical pyramid compare to that of the cube?
10. If you had any prism and a pyramid with the same base and height, how would the volume of the pyramid compare to the prism?

As an added feature you may wish to construct a figure that folds in and out from one shape to several shapes as seen in the videos below.

• A rectangular prism in six parts (pdf) (doc) Answers (pdf) (doc)
• A cube in three parts (pdf) (doc) Answers (pdf) (doc)

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks