Angles in Tetrahedron and Octahedron

Today we have a question from one of the subscribers to Super Math Tips. Her question goes like this (I changed it a little to be more accurate):

How many angles can the center of a circumsphere around a tetrahedron and an octahedron (separated) form with the vetices of these polyhedra?

First, you should know what octahedron and tetrahedron are. Click on their names to find their pictures on Wikipedia.

And now, for the answer:
Let's start with the tetrahedron (the pyramid), since it's smaller and simpler.

A tetrahedron has four vertices. To form an angle, we need two points to connect to the center of the circumsphere. At first, we have a choice of 4 different points (vertices) to choose from. After we choose one vertex, we have only 3 choices left. So the number of angles is seemingly 4 * 3 = 12. However, when we choose like that, there is an importance to the order of selection, but in angles there is no importance. That is why we divide this number by 2 to get rid of duplicate angles (like vertices 1 and 2 and vertices 2 and 1, which form the same angle). So in a tetrahedron you can form 6 angles with the vertices.

Another way to solve this problem is using combinatorics. One of the functions in combinatorics allows us to choose r objects (in this case, vertices) out of n objects when order does not matter. It goes like this:
nCr = n!/r!(n - r)!
The exclamation mark means factorial.

We want to choose 2 vertices out of 4. We plug in the numbers, and we get 4C2 = 4!/2!2! = 6, the same answer.

The octahedron is solved the same way: it has six vertices, so we can form 6 * 5 / 2 = 15 angles, or we can use combinatorics to calculate 6C2 = 6!/2!4! = 15. However, this number may be wrong.

Three of these angles are straight lines. From top to bottom of the octahedron and the diagonals of the "middle square". If those straight lines count as 180 degree angles, we have 15 angles in an octahedron. If not, we need to subtract 3 angles, so we get 12 possible angles.

Another way to see that is at first we can choose from 6 vertices, but then we can choose only four: we can't choose the same vertex again and we can't choose the vertex opposite the one we chose. This gives us 6 * 4 / 2 = 12 angles.

Keep sending in questions!
Yours,
Nadav

nadavs