In general, a tetrahedron is a polyhedron with four sides.

If all faces are congruent, the tetrahedron is known as an isosceles tetrahedron. If all faces are congruent to an equilateral triangle, then the tetrahedron is known as a regular tetrahedron (although the term "tetrahedron" without further qualification is often used to mean "regular tetrahedron"). A tetrahedron having a trihedron all of the face angles of which are right angles is known as a trirectangular tetrahedron.

A general (not necessarily regular) tetrahedron, defined as a convex polyhedron consisting of four (not necessarily identical) triangular faces can be specified by its polyhedron vertices as (x_i,y_i,z_i), where i=1, ..., 4. Then the volume of the tetrahedron is given by

 V=1/(3!)|x_1 y_1 z_1 1; x_2 y_2 z_2 1; x_3 y_3 z_3 1; x_4 y_4 z_4 1|.
(1)

Specifying the tetrahedron by the three polyhedron edge vectors a, b, and c from a given polyhedron vertex, the volume is

 V=1/(3!)|a·(bxc)|.
(2)

If the edge between vertices i and j is of length d_(ij), then the volume V is given by the Cayley-Menger determinant

 288V^2=|0 1 1 1 1; 1 0 d_(12)^2 d_(13)^2 d_(14)^2; 1 d_(21)^2 0 d_(23)^2 d_(24)^2; 1 d_(31)^2 d_(32)^2 0 d_(34)^2; 1 d_(41)^2 d_(42)^2 d_(43)^2 0|.
(3)

Consider an arbitrary tetrahedron A_1A_2A_3A_4 with triangles T_1=DeltaA_2A_3A_4, T_2=DeltaA_1A_3A_4, T_3=DeltaA_1A_2A_4, and T_4=DeltaA_1A_2A_3. Let the areas of these triangles be s_1, s_2, s_3, and s_4, respectively, and denote the dihedral angle with respect to T_i and T_j for i!=j=1,2,3,4 by theta_(ij). Then the four face areas are connected by

 s_k^2=sum_(j!=k; 1<=j<=4)s_j^2-2sum_(i,j!=k; 1<=i,j<=4)s_is_jcostheta_(ij)
(4)

involving the six dihedral angles (Dostor; Lee). This is a generalization of the law of cosines to the tetrahedron. Furthermore, for any i!=j=1,2,3,4,

 V=2/(3l_(ij))s_is_jsintheta_(ij),
(5)

where l_(ij) is the length of the common edge of T_i and T_j (Lee 1997).

Given a right-angled tetrahedron with one apex where all the edges meet orthogonally and where the face opposite this apex is denoted s_k, then

 s_k^2=sum_(j!=k; 1<=j<=4)s_j^2.
(6)

This is a generalisation of Pythagoras's theorem which also applies to higher dimensional simplices (F. M. Jackson).

Let A be the set of edges of a tetrahedron and P(A) the power set of A. Write t^_ for the complement in A of an element t in P(A). Let F be the set of triples {x,y,z} in P(A) such that x,y,z span a face of the tetrahedron, and let G be the set of (e intersection f) union (e union f^_) in P(A), so that e,f in F and e!=f. In G, there are therefore three elements which are the pairs of opposite edges. Now define D, which associates to an edge x of length L the quantity (L/RadicalBox[1, 3]2)^2, p, which associates to an element t in P(A) the product of D(x) for all x in t, and s, which associates to t the sum of D(x) for all x in t. Then the volume of a tetrahedron is given by

 sqrt(sum_(t in G)(s(t^_)-s(t))p(t)-sum_(t in F)p(t))
(7)

(P. Kaeser).

The analog of Gauss's circle problem can be asked for tetrahedra: how many lattice points lie within a tetrahedron centered at the origin with a given inradius (Lehmer, Granville, Xu and Yau, Guy).

There are a number of interesting and unexpected theorems on the properties of general (i.e., not necessarily regular) tetrahedron (Altshiller-Court). If a plane divides two opposite edges of a tetrahedron in a given ratio, then it divides the volume of the tetrahedron in the same ratio (Altshiller-Court). It follows that any plane passing through a bimedian of a tetrahedron bisects the volume of the tetrahedron (Altshiller-Court).

Let the vertices of a tetrahedron be denoted A, B, C, and D, and denote the side lengths BC=a, CA=b, AB=c, DA=a^', DB=b^', and DC=c^'. Then if Delta denotes the area of the triangle with sides of lengths aa^', bb^', and cc^', the volume and circumradius of the tetrahedron are related by the beautiful formula

 6RV=Delta
(8)

(Crelle; von Staudt; Rouché and Comberousse; Altshiller-Court).

Let Delta_i be the area of the spherical triangle formed by the ith face of a tetrahedron in a sphere of radius R, and let epsilon_i be the angle subtended by edge i. Then

 sum_(i=1)^4Delta_i=[2(sum_(i=1)^6epsilon_i)-4pi]R^2,
(9)

as shown by J.-P. Gua de Malves. The above formula provides the means to calculate the solid angle subtended by the vertex of a regular tetrahedron by substituting epsilon_i=cos^(-1)(1/3) (the dihedral angle). Consequently,

 Omega=(Delta_i)/(R^2)=3cos^(-1)(1/3)-pi,
(10)

or approximately 0.55129 steradians.