be the area of the face opposite vertex Pi and let θij be the dihedral angle between the two faces of the tetrahedron adjacent to the edge PiPj. 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: Interior point 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 Fa, Fb, Fc, and Fd. Then For vertices A, B, C, and D, interior point P, and feet J, K, L, and M of the perpendiculars from P to the faces, and suppose the faces have equal areas, then Inradius Denoting the inradius of a tetrahedron as r and the inradii of its triangular faces as ri for i = 1, 2, 3, 4, we have with equality if and only if the tetrahedron is regular. If A1, A2, A3 and A4 denote the area of each faces, the value of r is given by . This formula is obtained from dividing the tetrahedron into four tetrahedra whose points are the three points of one of the original faces and the incenter. Since the four subtetrahedra fill the volume, we have . Circumradius Denote the circumradius of a tetrahedron as R. Let a, b, c be the lengths of the three edges that meet at a vertex, and A, B, C the length of the opposite edges. Let V be the volume of the tetrahedron. Then Circumcenter The circumcenter of a tetrahedron can be found as intersection of three bisector planes. A bisector plane is defined as the plane centered on, and orthogonal to an edge of the tetrahedron. With this definition, the circumcenter of a tetrahedron with vertices ,,, can be formulated as matrix-vector product: In contrast to the centroid, the circumcenter may not always lay on the inside of a tetrahedron. Analogously to an obtuse triangle, the circumcenter is outside of the object for an obtuse tetrahedron. Centroid The tetrahedron's center of mass computes as the arithmetic mean of its four vertices, see Centroid. Faces The sum of the areas of any three faces is greater than the area of the fourth face. Integer tetrahedra There exist tetrahedra having integer-valued edge lengths, face areas and volume.