The volume of a dodecahedron

I asked the boys what they would like to do for our Family Math project this morning and they said that they wanted to find the volume of a dodecahedron. This seemed like a pretty good follow up to yesterday’s project of finding the volume of an icosahedron – unfortunately this one is a little bit more difficult.

However, despite the difficulty, we had a great exploration and even had the nice surprise of finding an expression for the length of the red struts. To finish on a successful note, I had them find the volume of a cube by chopping it up into pyramids.

Here’s how the boys got started with the project – one difficulty that we encounter is finding the height of the pentagon pyramid, but not for lack of trying:

We took a break from filming for a bit to see if we could make any sense out of this pentagonal pyramid. We even tried to find the height by switching over to a triple-sized pyramid like we used yesterday. Eventually the boys found an approximation to the height that seemed to be pretty close. Even with that close approximation we have two problems:

(1) We don’t know how to find the area of a pentagon, and

(2) We don’t know the lengths of the red struts.

Our pyramid from yesterday helps with problem (2), and my younger son has a nice geometric idea that helps even more. That idea really illustrates how the Zometool set helps kids build geometric intuition.

Next up I let the boys implement my younger son’s idea from the last video. Now we can see a right triangle that will let us find the length of the red struts. Fun!

With our new idea in hand, we move to the whiteboard to do a little calculating. We find another fun surprise – the length of the triangle leg we are trying to find is a length that we already knew!

The next step was going to Wolfram Alpha to have it crunch the numbers for us. We’d already found the relationship between the long and medium sides in yesterday’s blog – knowing that relationship is a big help here!

Here’s the last step – finding the volume of a cube by chopping into pyramids. A much easier problem, but still an opportunity for some good geometric discussions:

So, a nice project despite not quite finding the volume we originally set out to find. It was a nice surprise to be able to find the length of the red struts, and it was really neat to hear my younger son’s idea about finding the height of an equilateral triangle. Fun morning.

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