Dear Uncle Colin,

I have an octahedron, and I’m not afraid to use it! But I am afraid to find the angle between two adjacent faces. How would you do that?

Protractors Lack Accuracy Tackling Octahedron

Hi, PLATO, and thanks for your message!

An octahedron is a shape made from eight equilateral triangles joined edge-to-edge; you can also see it as two square-based pyramids base-to-base, which is probably the simplest way to see the answer.

The angle between two edges is double the angle between a face and the base of the pyramid it’s part of - and that comes from a right-angled triangle!

This right-angled triangle is formed of the midpoint of a side, the centre-point of the pyramid’s base and the point of the pyramid. If the side length of the octahedron is two units, then the base of the triangle has length of 1 and hypotenuse of $\sqrt{3}$, so the cosine of the angle is $\frac{1}{\sqrt{3}}$.

However, that’s half of the angle we want! We can sort this out with a trig identity: $\cos(2\theta) = 2\cos^2(\theta) - 1$, which in this case is $\frac{2}{3} - 1$ or $-\frac{1}{3}$.

Whoosh

“Well, $\arcsin\left(\frac{1}{3}\right)$ is just under 20 degrees, about 19 and a half? So $\arccos\left(-\frac{1}{3}\right)$ is 20-odd degrees more than a right angle. I’ll call it 109.5.”

“Acceptable, sensei. Except for the units.”

Hope that helps!

- Uncle Colin