Every polyhedron, regular and nonregular, convex and concave, has a dihedral angle at every edge. A dihedral angle (also called the face angle) is the angle at which two adjacent faces meet. Every dihedral angle in a particular Platonic solid has the same value, called "the" dihedral angle. Thus, the dihedral angle of a cube is 90°, while the dihedral angle of a dodecahedron is 116° 34′.
"The" dihedral angle of each Platonic solid is:
|Name||exact dihedral angle (in radians)||approximate dihedral angle (in degrees)|
|Hexahedron or Cube||π/2||90°|
|Octahedron||π − arccos(1/3)||109.47°|
|Icosahedron||2·arctan(φ + 1)||138.19°|
where φ = (1 + √5)/2 is the golden mean.
- Analysis of the 5 Regular Polyhedra gives a step-by-step derivation of these exact values.