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A space-filling tetrahedral disphenoid inside a cube. Two edges have dihedral angles of 90°, and four edges have dihedral angles of 60°. A disphenoid is a tetrahedron with four congruent triangles as faces; the triangles necessarily have all angles acute. The regular tetrahedron is a special case of a disphenoid.
The dihedral angles for the edge-transitive polyhedra are: Picture Name Schläfli symbol Vertex/Face configuration exact dihedral angle ... Tetrahedron {3,3} (3.3.3)
The 6 edge lengths - associated to the six edges of the tetrahedron. The 12 face angles - there are three of them for each of the four faces of the tetrahedron. The 6 dihedral angles - associated to the six edges of the tetrahedron, since any two faces of the tetrahedron are connected by an edge.
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 ...
Given the edge length .The surface area of a truncated tetrahedron is the sum of 4 regular hexagons and 4 equilateral triangles' area, and its volume is: [2] =, =.. The dihedral angle of a truncated tetrahedron between triangle-to-hexagon is approximately 109.47°, and that between adjacent hexagonal faces is approximately 70.53°.
The dihedral angle of an elongated triangular bipyramid can be calculated by adding the angle of the tetrahedron and the triangular prism: [5] the dihedral angle of a tetrahedron between two adjacent triangular faces is arccos ( 1 3 ) ≈ 70.5 ∘ {\textstyle \arccos \left({\frac {1}{3}}\right)\approx 70.5^{\circ }} ;
The dihedral angles of the regular tetrahedron and regular octahedron are supplementary angles: they sum to . Omitting either the tetrahedron or the octahedron from these five angles produces a rational basis: there are no other rational relations between these angles. [ 7 ]
This dissection can also be seen within the runcic cubic honeycomb, with cube, tetrahedron, and rhombicuboctahedron cells. This dissection can be used to create a Stewart toroid with all regular faces by removing two square cupolae and the central cube. This excavated cube has 16 triangles, 12 squares, and 4 octagons. [1] [2]