Tetraëder

Tetraëder, also called tetrahedron in English, is a polyhedron with four faces. In its most regular form, the regular tetrahedron, all four faces are congruent equilateral triangles, and the solid has four vertices and six edges. It is the simplest of the Platonic solids and is self-dual, meaning its dual polyhedron is another tetrahedron of the same shape.

For a regular tetrahedron with edge length a, key measurements are as follows: the dihedral angle between any two adjacent faces is arccos(1/3) ≈ 70.53 degrees. The height from a vertex to the opposite face is h = √(2/3)·a ≈ 0.8165a. The volume is V = a^3/(6√2) ≈ 0.11785a^3, and the surface area is A = √3·a^2 ≈ 1.732a^2. The circumscribed sphere radius (distance from the center to a vertex) is R = a√6/4 ≈ 0.612a, and the inradius (distance from the center to a face) is r = a√6/12 ≈ 0.204a. The center of the circumscribed sphere coincides with the centroid of the vertices.

Symmetry-wise, the regular tetrahedron has a rotational symmetry group of order 12 (isomorphic to A4) and a full symmetry group of order 24 when reflections are included. Its geometry appears in various contexts, from molecular shapes (such as carbon in methane) to dice used in tabletop games (d4). The term derives from Greek tetra- “four” and hedra “face.”