Tonight my older son is at a school event. That gave me time to do a fun little extension of these two projects with my younger son.

First I reviewed the original problems:

My son solved the 2nd problem above by making triangles with sides 1, and . For this part of the project I wanted to show him a different triangle that has a side length of – a 30-60-90 triangle:

Now – for a little extra fun – we made a Zometool cube. That cube shows that the face diagonal (of a 1x1x1) cube has length . It also shows that the internal diagonal has length

Here’s the surprise – if we extend basically the same geometry to 4 dimensions, we find that the “long” internal diagonal of a 1x1x1x1 cube has length 2, and that there’s a secret little 30-60-90 triangle hiding in the cube!