Trigonometry of a tetrahedron
The trigonometry of a tetrahedron explains the relationships between the lengths and various types of angles of a general tetrahedron. Trigonometric quantities. Classical trigonometric quantities. The following are trigonometric quantities generally associated to a general tetrahedron: Let formula_1 be a general tetrahedron, where formula_2 are arbitrary points in three-dimensional space. Furthermore, let formula_3 be the edge that joins formula_4 and formula_5 and let formula_6 be the face of the tetrahedron opposite the point formula_4; in other words: where formula_10 and formula_11. Define the following quantities: Area and volume. Let formula_21 be the area of the  face formula_22. Such area may be calculated by Heron's formula (if all three edge lengths are known): or by the following formula (if an angle and two corresponding edges are known): Let formula_25 be the altitude from the point formula_4 to the face formula_6. The volume formula_28 of the tetrahedron formula_29 is given by the following formula: formula_30It satisfies the following relation: where formula_32 are the quadrances (length squared) of the edges. Basic statements of trigonometry. Affine triangle. Take the face formula_6; the edges will have lengths formula_34 and the respective opposite angles are given by formula_35. The usual  laws for planar trigonometry of a triangle hold for this triangle. Projective triangle. Consider the projective (spherical) triangle at the point formula_15; the vertices of this projective triangle are the three lines that join formula_15 with the other three vertices of the tetrahedron. The edges will have spherical lengths formula_38 and the respective opposite spherical angles are given by formula_39. The usual laws for spherical trigonometry hold for this projective triangle. Laws of trigonometry for the tetrahedron. Alternating sines theorem. Take the tetrahedron formula_40, and consider the point formula_15 as an apex. The Alternating sines theorem is given by the  following identity:formula_42One may view the two sides of this identity as corresponding to clockwise and counterclockwise orientations of the surface. The space of all shapes of tetrahedra. Putting any of the four vertices in the role of "O" yields four such identities, but at most three of them are independent; if the "clockwise" sides of three of the four identities are multiplied and the product is inferred to be equal to the product of the "counterclockwise" sides of the same three identities, and then common factors are cancelled from both sides, the result is the fourth identity. Three angles  are the angles of some triangle if and only if their sum is 180° (π radians). What condition on 12 angles is necessary and sufficient for them to be the 12 angles of some tetrahedron? Clearly the sum of the angles of any side of the tetrahedron must be 180°. Since there are four such triangles, there are four such constraints on sums of angles, and the number of degrees of freedom is thereby reduced from 12 to 8. The four relations given by the sine law further reduce the number of degrees of freedom, from 8 down  to not 4 but 5, since the fourth constraint is not independent of the first three. Thus the space of all shapes of tetrahedra is 5-dimensional. Law of sines for the tetrahedron. See: Law of sines 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: 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. 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 of the tetrahedron, as follows: formula_53 Skew distances between edges of tetrahedron. Take the general tetrahedron formula_40 and let formula_55 be the point on the edge formula_3 and formula_57 be the point on the edge formula_58 such that the line segment formula_59  is perpendicular to both formula_3 & formula_58. Let formula_62 be the length of the line segment formula_59. To find formula_62: First, construct a line through formula_65 parallel to formula_66 and another line through formula_4 parallel to formula_58. Let formula_69 be the intersection of these two lines. Join the points formula_69 and formula_5. By construction, formula_72 is a parallelogram and thus formula_73 and formula_74 are congruent triangles. Thus, the tetrahedron formula_40 and formula_76 are equal in volume. As a consequence, the quantity formula_62 is equal to the altitude from the point formula_65 to the face formula_79 of the tetrahedron formula_80; this is  shown by translation of the line segment formula_59. By the volume formula, the tetrahedron formula_80 satisfies the following relation: formula_83where formula_84 is the area of the triangle formula_79. Since the length of the line segment formula_86 is equal to formula_87 (as formula_72 is a parallelogram): formula_89where formula_90. Thus, the previous relation becomes: formula_91To obtain formula_92, consider two spherical triangles: Combining the two equations gives the following result:formula_104 Making formula_105 the subject:formula_106Thus, using the cosine law and some basic trigonometry:formula_107Thus:formula_108So:formula_109formula_110 and formula_111 are obtained by permutation of the edge lengths. Note that the denominator is a  re-formulation of the Bretschneider-von Staudt formula, which evaluates the area of a general convex quadrilateral.