Tetrahedra

A tetrahedron is a convex polyhedron with four triangular faces, six edges, and four vertices. If its four faces are congruent equilateral triangles, the figure is a regular tetrahedron, one of the five Platonic solids. Tetrahedra are the simplest three-dimensional solids and can be studied in Euclidean geometry as well as in higher dimensions.

In a tetrahedron, each vertex is the meeting point of three edges, and the figure is self-dual, meaning its vertices correspond to faces in the dual polyhedron. The Euler characteristic is V − E + F = 2, with V = 4, E = 6, and F = 4. The dihedral angle between any two adjacent faces in a regular tetrahedron is arccos(1/3), about 70.53 degrees. The rotational symmetry group of a regular tetrahedron is isomorphic to A4 (order 12), and the full symmetry group (including reflections) is isomorphic to S4 (order 24).

For a regular tetrahedron with edge length a, the volume is V = a^3/(6√2), and the surface area is A = √3 a^2. The circumradius is R = a√6/4 and the inradius is r = a√6/12; the height from a vertex to the opposite face is h = √(2/3) a.

Beyond the regular form, tetrahedra can be irregular, with unequal faces. A disphenoid (or equifacial tetrahedron) has four congruent triangular faces, while a right tetrahedron has three mutually perpendicular edges meeting at a vertex. Tetrahedra appear in molecular geometry, computer graphics, finite element methods, and crystallography, where their properties underpin triangulation and modeling.