How calculate the FOV of the faces of a dodecahedron

Question

Having a dodecahedron I want to calculate the field of view (FOV) in degrees of the faces of the shape. If each pentagon has a circunscribed circle, I want to know or how to calculate the angle from the center of the dodecahedron to both sides of the circle

Answer 1

The radius of the circumscribing circle on your diagram can be found from the edge length of the pentagon $a$ and is $r=\frac{acosec(\pi/5)}{2}$

The radius R of the spherical surface passing through all 20 identical vertices of a dodecahedron is given by

${R=\frac{\sqrt{3}(\sqrt{5}+1)a}{4}}$

(see also What is the height of a Dodecahedron in terms of it's edge length and can the formula be simplified or generalised)

Then if $\alpha$ is half the angle you require $sin\alpha = \frac{r}{R}$

The answer is approximately 1.3047 rads or 74.755 degrees