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exact dihedral angle (radians) dihedral angle – exact in bold, else approximate (degrees) Platonic solids (regular convex) Tetrahedron {3,3} (3.3.3) arccos ( 1 / 3 ) 70.529° Hexahedron or Cube {4,3} (4.4.4) arccos (0) = π / 2 90° Octahedron {3,4} (3.3.3.3) arccos (- 1 / 3 ) 109.471° Dodecahedron {5,3} (5.5.5) arccos ...
An angle of 0° means the face normal vectors are antiparallel and the faces overlap each other, which implies that it is part of a degenerate polyhedron. An angle of 180° means the faces are parallel, as in a tiling. An angle greater than 180° exists on concave portions of a polyhedron. Every dihedral angle in an edge-transitive polyhedron ...
The rhombic dodecahedron is a polyhedron with twelve rhombi, each of which long face-diagonal length is exactly times the short face-diagonal length [1] and the acute angle measurement is (/). Its dihedral angle between two rhombi is 120°. [2]
This fact can be used to calculate the dihedral angles themselves for a regular or edge-symmetric ideal polyhedron (in which all these angles are equal), by counting how many edges meet at each vertex: an ideal regular tetrahedron, cube or dodecahedron, with three edges per vertex, has dihedral angles = / = (), an ideal regular octahedron or ...
The dihedral angle of a triangular cupola between square-to-triangle is approximately 125°, that between square-to-hexagon is 54.7°, and that between triangle-to-hexagon is 70.5°. Therefore, the dihedral angle of a cuboctahedron between square-to-triangle, on the edge where the base of two triangular cupolas are attached is 54.7° + 70.5 ...
the dihedral angle of a rhombicuboctahedron between two adjacent squares on both the top and bottom is that of a square cupola 135°. The dihedral angle of an octagonal prism between two adjacent squares is the internal angle of a regular octagon 135°. The dihedral angle between two adjacent squares on the edge where a square cupola is ...
The dihedral angle equals (+). The ratio between the lengths of the long edges and the short ones equals +, which is the golden ratio. Part of each face lies inside the solid, hence is invisible in solid models.
The dihedral angle equals (/ ()). Note that the face centers of the snub dodecahedron cannot serve directly as vertices of the pentagonal hexecontahedron: the four triangle centers lie in one plane but the pentagon center does not; it needs to be radially pushed out to make it coplanar with the triangle centers.