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°. It is not 60°, even though every face is an equilateral triangle, because the faces tilt away from each other in three dimensions rather than meeting in a plane. Anyone cutting mitres for a tetrahedral frame needs 70.53° and not the 60° the flat drawing suggests.

The lopsided centre

Here is the property that surprises people. The centre of a tetrahedron is not halfway up. The inradius is a√6/12 and the height is a√6/3, so the inscribed sphere touches the base at exactly one quarter of the height. Slice a tetrahedron in half by volume and the cut sits at about 0.206 of the height from the base, not 0.5, because almost all the material is down at the wide end.

That asymmetry is why a tetrahedral die is such an awkward object. It has no face opposite a vertex to land upward, which is why a d4 is read from the numbers along its bottom edges rather than from a top face.

Packing

Tetrahedra do not tile space, which is a genuinely surprising fact given that they are the simplest solid. Aristotle claimed they did and was believed for roughly 1,800 years before the gap was measured: five tetrahedra around a shared edge leave 7.36° unfilled, since 360 divided by the 70.53° dihedral angle is 5.10 rather than a whole number. Space fills instead with alternating tetrahedra and octahedra, which is the structure behind every space frame roof you have stood under.

For unit edge (a = 1): V ≈ 0.11785, SA ≈ 1.73205, h ≈ 0.81650, inradius ≈ 0.20412, circumradius ≈ 0.61237.