tetrahedrons

A tetrahedron is a polyhedron composed of four triangular faces, six edges, and four vertices. It is the simplest form of a polyhedron in three-dimensional space and serves as the three-dimensional analogue of a triangle, a basic example of a simplex in Euclidean geometry. Any four non-coplanar points determine a tetrahedron.

The regular tetrahedron has four congruent equilateral triangle faces and exhibits high symmetry. All vertices are equidistant from the center, and its rotational symmetry group has order 12 (24 including reflections). The regular tetrahedron is self-dual, meaning its dual polyhedron is another tetrahedron of the same shape.

Key measurements include edge length a. The volume is V = a^3/(6√2) and the surface area is A = √3 a^2. For the regular tetrahedron, the inradius is r = a√6/12 and the circumradius is R = a√6/4, with the centroid at the intersection of the medians.

Beyond the regular form, there are general tetrahedra with arbitrary edge lengths and face shapes. A disphenoid (isosceles tetrahedron) has four congruent triangular faces but is not necessarily regular. The Euler relation V − E + F = 2 holds for any convex tetrahedron, where V = 4, E = 6, and F = 4.

Tetrahedra appear in chemistry and crystallography as the basis for molecular geometries such as methane, and in computer graphics and numerical methods as simplices for meshes and interpolation.