Angle dodecahedron makes when vertice "equipositioned" against plane

Dec 2017
Melbourne Australia
Hi all

I don't know if this is technically differential geometry but I have a problem I cannot solve. (Crying)

Maybe you can help me.

My words are probably not technically correct but I will try to explain what I am after as best I can.

I am making a dodecahedron and I want to know the angle the three edges make when a vertex is positioned "symmetrically perpendicular" to a plane.

That is the line made from the vertex positioned on the plane to the "opposite" vertex in the dodecahedron is at perpendicular to the said plane.

Another way of explaining this is to imagine a dodecahedron spinning "symmetrically" on a vertex. What is the angle the three edges converging at the vertex make with the plane? It will be one angle as it is spinning "symmetrically".

Many thanks to you all.