# 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 gives a step-by-step derivation of these exact values.