Exploring some special triangles with Zometool

Yesterday I started in on the “special triangles” section of our Prealgebra book with my younger son. I also happen to be studying some simple 3-dimensional figures with my older son. Since we were going on a little trip today I thought it would be fun to try to combine the two topics into a fun little Zometool project.

One difficulty is that (I’m pretty sure) you can’t make a 30-60-90 triangle with the Zometool struts. No matter. If you use blue struts for the 1 and 2 lengths, you can use the yellow struts for the \sqrt{3} length even if they don’t plug in properly.

We started by looking at two simple triangles – a 45-45-90 triangle and a 30-60-90 triangle. After the kids finished building them I asked them what they noticed:

The next challenge was to build and study a tetrahedron. We found that the height of the pyramid would be yellow struts, but we couldn’t find the right length. In this project we tried to see if we could calculate that height. This part of our project combines 3D geometry, 30-60-90 triangles, and the Pythagorean theorem:

The last part of the project was going through the calculation of both the height and the volume of our tetrahedron. Understanding the 3d geometry was a tiny bit confusing for my younger son, but the Zometool set definitely helped him see the geometry a little better. I think that if we only had the picture on the whiteboard he would have really struggled to calculate the height of the pyramid.

The last part of this project was also a nice way to get a little practice manipulating the square roots that come up all over the place in the height and volume calculations.

So, a fun impromptu project where the Zometool set helps us get our arms around both 2d and 3d geometry. Along the way we got a little practice calculating with square roots and even figured out the volume of a tetrahedron made out of green Zometool struts. It is nice little coincidence that the boys happen to be studying similar subjects right now 🙂