# Dihedral Angle Calculator for Polyhedra

If three faces meet at a vertex on a polyhedron and you know the three vertex angles, then you can use formulas from spherical trigonometry to compute the angles between two adjacent faces, the dihedral angle.

In the figure below, the vertex angles are labeled *a*, *b*, and *c*, while the dihedral angles are labled *A*, *B*, and *C*. (In spherical trigonometry, the subtending angles of a solid angle correspond to *a*, *b*, and *c*, while the surface angles correspond to *A*, *B*, and *C*.)

You can use the calculator on the left to compute the dihedral angles for any degree-three vertex, for example, in a tetrahedron. Some dihedral angles of common polyhedra are given in the table below.

Polyhedron | Exact Dihedral Angle | Dihedral Angle in Degrees | |

P L A T O N I C | Regular Tetrahedron | cos^{-1}(1/3) | 70.52878° |

Cube | 90° | 90° | |

Octahedron | cos^{-1}(-1/3) | 109.47122° | |

Dodecahedron | cos^{-1}(-1/√5) | 116.56505° | |

Icosahedron | cos^{-1}(-√5/3) | 138.18969° | |

Cuboctahedron | cos^{-1}(-1/√3) | 125.26439° | |

Rhombic Dodecahedron | 120° | 120° | |

Rhombic Triacontahedron | 144° | 144° |

© *Had2Know 2010
*