Dihedral angle
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In aerospace engineering, the dihedral is the angle between the two wings; see dihedral.
In geometry, the angle between two planes is called their dihedral angle. It can be defined as the angle between two lines normal to the planes.
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) |
|---|---|---|
| Tetrahedron | arccos(1/3) | 70.53° |
| Hexahedron or Cube | π/2 | 90° |
| Octahedron | π − arccos(1/3) | 109.47° |
| Dodecahedron | 2·arctan(φ) | 116.56° |
| Icosahedron | 2·arctan(φ + 1) | 138.19° |
where φ = (1 + √5)/2 is the golden mean.
External links
- Analysis of the 5 Regular Polyhedra (http://kjmaclean.com/Geometry/Platonic.html) gives a step-by-step derivation of these exact values.Template:Geometry-stub