How many faces does a tetrahedron have?
Trigonometry of a tetrahedron relates the areas of each face of the tetrahedron and the dihedral angles about a point. It is given by the following identity: Take the general tetrahedron formula_40 and project the faces formula_45 onto the plane with the face formula_46. Let formula_47. Then the area of the face formula_46 is given by the sum of the projected areas, as follows:formula_49By substitution of formula_10 with each of the four faces of the tetrahedron, one obtains the following homogeneous system of linear equations:formula_51This homogeneous system will have solutions precisely when: formula_52By expanding this determinant, one obtains the relationship between the dihedral angles
Tetrahedron tetrahedron. Let Δ be the area of the face opposite vertex "P" and let "θ" be the dihedral angle between the two faces of the tetrahedron adjacent to the edge "PP". The law of cosines for this tetrahedron, which relates the areas of the faces of the tetrahedron to the dihedral angles about a vertex, is given by the following relation: Let "P" be any interior point of a tetrahedron of volume "V" for which the vertices are "A", "B", "C", and "D", and for which the areas of the opposite faces are "F", "F", "F", and "F". Then For