Making a lamp
How can one accurately calculate the dihedral angle? Fortunately, I have a book by Peter R. Cromwell, Polyhedra¹, that explains Euclid's strategy for creating a dodecahedron. Basically, Euclid started with a cube, and then added a roof-shaped structure to each face of the cube. Taken together, the faces of the roofs create the pentagons.
So now you have a bunch of roofs. If you can figure out the pitch of one of those roofs, you're in business. But how? Cromwell tells us that the pitch of the roof is based on the golden ratio, which makes sense given that a dodecahedron can be formed around three intersecting golden rectangles.
(1) Peter R. Cromwell, Polyhedra. Cambridge University Press, Cambridge, 1997.