Regular Tetrahedron Calculator

A regular tetrahedron is the simplest Platonic solid: four equilateral triangles, four vertices, every edge the same length. The formulas are unusually clean because of the high symmetry.

Given edge length a:

Volume: V = a³√2 / 12

Surface area: SA = a²√3 (four equilateral triangles, each with area a²√3/4)

Height from vertex to opposite face: h = a√6 / 3

Inradius (inscribed sphere): rᵢ = a√6 / 12

Circumradius (sphere through all vertices): R = a√6 / 4

The circumradius is exactly three times the inradius. This 3:1 ratio holds for all regular tetrahedra regardless of size.

Deriving the volume

Start with the base equilateral triangle of side a: area = a²√3/4. The centroid of the base is at distance a/√3 from each vertex. The height from apex to base satisfies h² + (a/√3)² = a², giving h = a√(2/3) = a√6/3. Volume = (1/3)·base·height = a³√2/12.

Where it appears in practice

Carbon chemistry: sp3 hybridization gives methane (CH₄) its tetrahedral shape, with bond angles of arccos(−1/3) ≈ 109.47°.

Crystal structures: the diamond lattice consists of two face-centered cubic sublattices shifted so each atom sits at the center of a tetrahedron formed by its four nearest neighbors.

Finite element analysis: arbitrary 3D volumes are decomposed into tetrahedra because the geometry is the simplest possible for computing spatial gradients.

The dihedral angle between two faces is arccos(1/3) ≈ 70.53°. This is not 60° even though the faces are equilateral triangles, because the faces tilt relative to each other in 3D space.

For unit edge (a = 1): V ≈ 0.11785, SA ≈ 1.73205, h ≈ 0.81650.