My younger son is starting to learn about coordinates in 3 dimensions. I thought that spending a little time finding the coordinates of the corners of a tetrahedron and an octahedron would make for a nice project this morning.
We started with the tetrahedron and found the coordinates for the bottom face. Once nice thing about the discussion here was talking about the various choices we had for how to look at the tetrahedron:
Having found the coordinates for the bottom face, we now moved on to finding the coordinates for the top vertex:
Now we moved on to trying to find the coordinates for the corners of the octahedron. Here the choices for how to orient the object are a little more difficult:
Finally, we talked through how we would find the coordinates of the octahedron if we had it oriented in a different way. This was a good discussion, but was also something that confused the boys a bit more than I thought. We spent about 10 min after the project talking through how to find the height. Hopefully the discussion here helps show why this problem is a pretty difficult one for kids:
Mike's Math Page 
Tetrahedron: (0,0,0), (0,1,1), (1,0,1), (1,1,0).
Octahedron: (+/-1,0,0), (0,+/-1,0), (0,0,+/-1).
A friend and I had to work with a tetrahedron for a physics homework, and had all these sqrt(3)s in it and so on. When the HW solutions came out and used the easier orientation above, he exclaimed “I would have killed for coordinates like that!”