Images from a lesson on the volume of regular polyhedra

How it might work – in a nutshell (maybe a coconut)

Working as a class, students will construct the 50 pyramids that make up the five Platonic Solids. These pyramids all have a height of 5 units, and each one has a polygon that is a face of one of the solids as its base. The pyramids completely fill the volume of the solid, and their apexes meet at the center of the solid.

Students will have time to explore the pyramids, looking for unifying or distinguishing features. The apothem and side length of each polygonal base are given, so students just need height to calculate volume. If they don’t discover that they are all the same height, lead them a little! Depending on the level of your students, they could measure the given lengths (non-standard units) and use proportions; they could use the Pythagorean Theorem, since they are given one leg and the hypotenuse of the triangle whose other leg is the height; they could measure the dihedral angle, take half, and use trig; or you can tell them! In a course beyond elementary geometry, you might not want to give them any measurements.

Students will sort themselves into groups according to the size and shape of the pyramids they’ve built, and then assemble the solids. I recommend small masking tape loops b/c they are quite easily removable. I made sure that one of the pyramids was folded so that the printing is outside, but that is not a biggie if you’ve recorded the measurements.

More time for sharing observations, followed by finding the volumes of the whole solid, which now are simple multiples of the volumes of the pyramids. Have students graph the values: number of faces vs. volume. Solicit observations regarding patterns, trends, limits! Does it look familiar? For my 10th graders, “As the number of faces increases, the volume decreases, but not at a constant rate,” is probably a good answer.

The next day, introduce the plastic “cases” for the solids. Ask the groups to reassemble their pyramids in the cases, and then empty them again. Introduce a sphere to the problem (for my nets, an 80 mm decapitated plastic Christmas ornament was perfect – and I had 5 of them!). Ask for which case(s) would it be too big, too small, or just right (fits but doesn’t wiggle). Have students try to fit it – wait for surprised reactions!

Re-play the Russell Kightley animation at http://youtu.be/DeNv6YCoXTw to remind students of surface area of a sphere. Point out the pyramid that he’s drawn inside. Watch the “Explanation of the Volume of a Sphere Formula” video by mathematicsonline at http://youtu.be/xuPl_8o_j7k. Point out all of the pyramids! (Generating the formula can also be done as a class discussion, but I like these 2 videos together.) Reinforce the formula and then calculate the volume of the sphere with height 5 units. Have students plot this value on their graphs. Introduce the term asymptote. Smile!

whew! (really big coconut)

Leave a comment if you would like the nets!