Exploring angles in platonic solids

The boys wanted to do a project with the Zometool set this morning, so I had them flip through Zome Geometry to find a project. They found a neat one about measuring dihedral angles in the platonic solids.

This project is a really great example of how useful the Zometool set is for exploring 3D geometry. Without the Zometool set this would be a pretty challenging project for kids. With it – we can build the shapes from scratch, measure the angles with a protractor, and – as an extra bonus – have every thing cleaned up and put away in an hour!

We started by building the shapes off camera and then having the kids talk about the project and talk about the shapes. Their initial thoughts about the various angles in the shapes hint at some of the ideas that they will learn about in this project:


First up, we explore the dihedral angle in a tetrahedron. It takes a while for the boys to figure out how to measure this angle (and even what angle it is). Once they see what the angle is, figuring out how to use the protractor to measure that angle takes a little bit of time. Luckily, though, once we solve this measuring problem for the tetrahedron, we can use the same idea to measure the dihedral angles in the remaining shapes.


Next up is measuring the dihedral angle in a cube. After the challenge of finding the correct angle in the tetrahedron, it is probably a lucky break that the next shape we looked at was fairly easy.


Next up was the octahedron. While holding the shape in his hands, my older son thinks that there is more than one angle we need to measure. He wondered about these angles at the beginning of the project, too. Luckily the Zometool set helps him see that, in fact, all of the dihedral angles are the same. Yet another educational win for the Zometool set!


The dodecahedron is a little larger than the other shapes we’ve looked at. I try to help out a little bit at the beginning, but wisely the boys ignore my help – always the right decision on any engineering problem!

For this shape we spent a little bit of time comparing the angles in the pentagons with the dihedral angles, too.


It was probably lucky that the icosahedron was the last shape we looked at. In retrospect I should have had the boys make this shape with side lengths equal to two long blue struts. The problem with the size of our shape was that the protractor was too big to fit inside the triangles to get a good measure of the angle. It took a little bit of playing around, but they were able to see that the angle was about 140 degrees.

We wrapped up this video with a quick review of what they learned during the project.


So a fun little geometry project. I’m really happy that we can use the Zometool set to introduce some basic ideas from 3D geometry to the kids. Hopefully seeing ideas like these one early on will take some of the mystery out of learning more 3D geometry when they encounter it later in school.