pentagon is the shape of each face of a tetrahedron.
what is the shape of each face of a tetrahedron?
pentagon
topologically, as a near-miss johnson solid, the four hexagons corresponding to the face planes of a tetrahedron are triambi, with equal edges but alternating angles, while the pentagons only have reflection symmetry. 





























section::::full truncation.





























if all of a triakis tetrahedron's vertices, of both kinds, are truncated, the resulting solid is an irregular icosahedron, whose dual is a "trihexakis truncated tetrahedron".





























truncation of only the 3-valence vertices yields the order-3 truncated triakis tetrahedron, which looks like a tetrahedron with each face raised by a low triangular frustum. the dual to that truncation will be the triakis truncated tetrahedron.
truncated triakis tetrahedron
section::::laws of trigonometry for the tetrahedron.:law of sines for the tetrahedron.





























see: law of sines





























section::::laws of trigonometry for the tetrahedron.:law of cosines for the tetrahedron.





























the law of cosines for the tetrahedron relates the areas of each face of the tetrahedron and the dihedral angles about a point. it is given by the following identity:





























section::::laws of trigonometry for the tetrahedron.:relationship between dihedral angles of tetrahedron.





























take the general tetrahedron formula_40 and project the faces formula_45 onto the plane with the face formula_46. let formula_47.
trigonometry of a tetrahedron