“Can we go buy some cheese?” I ask. “But we already have cheese in the house!” my dad replies. “Yeah but… I need a new block of cheese,” I insist. “It’s for a math project!”
“Oh,” my dad acknowledges. He knows never to question my pursuits when the word “math” is involved.
Remember how in my last post, I was trying to find the volume of the funky polyhedron ABCDEF…
Well, today, I’m going to use this block of cheese to create the hexagonal prism, then demonstrate the process of finding the volume through physically cutting chunks off of the hexagonal prism until I create ABCDEF, the funky polyhedron in question.
Okay. First, look at the cheese from a bird’s-eye-view.
Place a hexagon on top of it, and cut along the sides of the hexagon.
We are trying to create a hexagonal prism, like so:
Take the paper hexagon, and fold it into a triangle, and trace that triangle on the cheese surface. I used my knife to “trace” (or should I say carve…)
Okay. Flip the cheese prism onto its side…
and make these marks on its side faces, like so:
Flip it over and keep doing that until you’ve “zig-zagged” around its perimeter
See how your marks form a triangle? That triangle is the base of a tetrahedron… let’s cut along those marks to remove that tetrahedron.
Go around and cut off two more tetrahedra for three in total. Notice how all three tetrahedra are the same size. Symmetry is a beautiful thing.
Now, flip the funky polyhedron over so that the second hexagonal face is facing up. Mark a triangle on the hexagonal face like so.
Begin to cut off more tetrahedra, just like we did before.
When you’re done, you should be left with one funky-shaped polyhedron and six small congruent tetrahedra.
Remember that we are trying to find the volume of the funky polyhedron. We can do that by subtracting the volume of the six small tetrahedra from the original hexagonal prism. So let’s look at a tetrahedron…
Its height makes a right angle with the piece of paper, so this is a right tetrahedron, which will make our volume calculations easier. First, let’s trace out its base…
Note how its base is part of the paper hexagon we used to cut our prism in the beginning!
Let the side length of the hexagon be a. The triangular base is made of two smaller 30-60-90 triangles, so we can find its areas as follows:
Now we look at the height of the tetrahedron. Since it’s a right tetrahedron its height is just the length of the pink edge.
Remember what we got for the area of the base?
And we use that value to help us calculate the volume of the tetrahedra, like so:
Now, we look at everything all together: (volume funky solid) + 6*(volume tetrahedra) = (volume original hexagonal prism).
It’s pretty straightforward to calculate the volume of the original hexagonal prism: just (base) * (height).
Now calculate the total volume of the tetrahedra:
We are so close! One final step, just subtract the 6 tetrahedra volume from the original hexagonal prism volume!
All together now…
And there you have it! For any “funky polyhedron” truncated from a hexagonal prism like the one above, we have
Volume = a2h √3,
where a is the side length of the prism base and h is the height.
(Note that there is a √3 in the answer because the base of the prism has six sides. What if the base had n sides? Could you find a formula to generalize for some arbitrary number of sides n and height h?)
For the original problem, I found a to be 2√3 and h to be (√3)(3+√5), so my answer came out to be 36(3+√5) = 108+36√5. (You can find the details of my calculations here.)
I hope this post was helpful and made the process (and motivation behind the process) easier to visualize and comprehend. Math is truly so powerful and I love how the simplest techniques can be utilized in the most creative of ways.
I feel very accomplished now because I can now say that I spent an afternoon chopping up cheese blocks in the pursuit of geometry. Euclid and Pythagoras would be very proud…