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Tetrahedron


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 1905, pp. 252-293; Lee 1997). 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, pers. comm., Feb. 20, 2006).

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, pers. comm.).

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 1940, Granville 1991, Xu and Yau 1992, Guy 1994).

There are a number of interesting and unexpected theorems on the properties of general (i.e., not necessarily regular) tetrahedron (Altshiller-Court 1979). 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 1979, p. 89). It follows that any plane passing through a bimedian of a tetrahedron bisects the volume of the tetrahedron (Altshiller-Court 1979, p. 90).

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 1821, p. 117; von Staudt 1860; Rouché and Comberousse 1922, pp. 568-576 and 643-664; Altshiller-Court 1979, p. 249).

Let Delta_i be the area of the spherical triangle formed by the ith face of a tetrahedron circumscribed 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 around 1740 or 1783 (Hopf 1940). The above formula provides the means to calculate the solid angle Omega subtended from a vertex by the opposite face of a regular tetrahedron by substituting epsilon_i=cos^(-1)(1/3) (the dihedral angle) into the above formula. Consequently,

Omega=(Delta_i)/(R^2)=3cos^(-1)(1/3)-pi
(10)
=cos^(-1)((23)/(27)),
(11)

or approximately 0.55129 steradians.


See also

Disphenoid, Isosceles Tetrahedron, Regular Tetrahedron Explore this topic in the MathWorld classroom

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References

Altshiller-Court, N. "The Tetrahedron." Ch. 4 in Modern Pure Solid Geometry. New York: Chelsea, pp. 48-110 and 250, 1979.Balliccioni, A. Coordonnées barycentriques et géométrie. Claude Hermant, 1964.Couderc, P. and Balliccioni, A. Premier livre du tétraèdre à l'usage des éléves de première, de mathématiques, des candidats aux grandes écoles et à l'agrégation. Paris: Gauthier-Villars, 1935.Crelle, A. L. "Einige Bemerkungen über die dreiseitige Pyramide." Sammlung mathematischer Aufsätze u. Bemerkungen 1, 105-132, 1821.Dostor, G. Eléments de la théorie des déterminants, avec application à l'algèbre, la trigonométrie et la géométrie analytique dans le plan et l'espace, 2ème ed. Paris: Gauthier-Villars, pp. 252-293, 1905.Gardner, M. "Tetrahedrons." Ch. 19 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 183-194, 1984.Geometry Technologies. "Tetrahedron." http://www.scienceu.com/geometry/facts/solids/tetra.html.Granville, A. "The Lattice Points of an n-Dimensional Tetrahedron." Aequationes Math. 41, 234-241, 1991.Guy, R. K. "Gauß's Lattice Point Problem." §F1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 240-241, 1994.Hopf, H. "Selected Chapters of Geometry." ETH Zürich lecture, pp. 1-2, 1940. http://www.math.cornell.edu/~hatcher/Other/hopf-samelson.pdf.Lee, J. R. "The Law of Cosines in a Tetrahedron." J. Korea Soc. Math. Ed. Ser. B: Pure Appl. Math. 4, 1-6, 1997.Lehmer, D. H. "The Lattice Points of an n-Dimensional Tetrahedron." Duke Math. J. 7, 341-353, 1940.Rouché, E. and de Comberousse, C. Traité de Géométrie, nouv. éd., vol. 1: Géométrie plane. Paris: Gauthier-Villars, 1922.Rouché, E. and de Comberousse, C. Traité de Géométrie, nouv. éd., vol. 2: Géométrie dans l'espace. Paris: Gauthier-Villars, 1922.von Staudt, K. G. C. "Ueber einige geometrische Sätze." J. reine angew. Math. 57, 88-89, 1860.Xu, Y. and Yau, S. "A Sharp Estimate of the Number of Integral Points in a Tetrahedron." J. reine angew. Math. 423, 199-219, 1992.

Cite this as:

Weisstein, Eric W. "Tetrahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Tetrahedron.html

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