I liked the activity for a lot of reasons – it was a great way to review a little geometry and arithmetic, a nice opportunity to discuss more 3D geometry, and finally a lucky coincidence with our project from yesterday talking about the truncated cube and the truncated dodecahedron:

The unlucky part is that my younger son slept in because of daylight savings time – boo 😦 Oh well, here’s what I did with my older son:

Part 1: We first talked about the surface area and volume formulas for a sphere and then discussed how you could use the ratio in Honner’s post to define in an unusual way.

Once we had the idea that $\pi = (1/36) \frac{SA^3}{V^2}$ we extended that idea to other 3 dimensional shapes. The first shape was a cube.

Also, sorry for the bad camera shot initially on this one – didn’t realize I’d cut off the bottom of our whiteboard until mid way through the video – oops.

After the Cube discussion we moved on to a Tetrahedron. Since I wasn’t looking to study the tetrahedron too much today, we pulled the volume formula from Wikipedia.

Next we took a look at the two shapes from yesterday and tried to guess which one was more like a sphere:

Finally, we did the calculations for these to shapes to find out what “ was for the truncated cube and the truncated dodecahedron according to Patrick Honner’s formula:

I left some of the calculations at the end of the video as an exercise for my son. After he finished the calculations he wanted to try to extend the idea to 4 dimensions – yes!! Sadly this idea proved to be a little harder than he expected, but maybe we’ll look at it tonight.

## Comments

It’s so much fun to watch you and your sons bring these ideas to life and then take them in different directions. Thanks for sharing.