Tetrahedron Volume Calculator (regular edge or four vertex coordinates)

Understanding the tetrahedron volume calculator — concept and principle

This calculator finds the volume of a tetrahedron, either a regular one from its edge length or any tetrahedron from the coordinates of its four corners. For a regular tetrahedron it also gives the surface area, height and the radii of the inscribed and circumscribed spheres; for four vertices it adds face areas, height and centres. Use it for solid geometry, 3D modelling, mesh analysis, chemistry (tetrahedral molecules) and vector calculus.

Four equilateral faces; the volume a³ ÷ (6√2) equals ⅓ × base area × height.

How it is calculated

Regular tetrahedron with edge a:

Volume V = a³ ÷ (6√2); surface area √3·a²; height a·√(2/3).
Circumradius a·√6 ÷ 4; inradius a ÷ √24.

Any tetrahedron with corners P₀…P₃:

u = P₁ − P₀, v = P₂ − P₀, w = P₃ − P₀; V = |u · (v × w)| ÷ 6.
Each face area is half the length of a cross product; height from P₃ = 3V ÷ area of face P₀P₁P₂; inradius = 3V ÷ total surface area.

If the four points lie in one plane, V = 0 and there is no tetrahedron.

The principle

The scalar triple product u · (v × w) gives the volume of the parallelepiped spanned by three edge vectors. A tetrahedron is a pyramid with a triangular base: its volume is one third of base area × height, and its base is half the parallelogram, so it is one sixth of the parallelepiped. For a regular tetrahedron, the base is an equilateral triangle of area (√3/4)a² and the height is a√(2/3); one third of their product simplifies to a³ ÷ (6√2).

Worked example

Default inputs: regular tetrahedron, edge a = 6.

Volume = 216 ÷ (6√2) = 36 ÷ √2 ≈ 25.4558.
Surface area = √3 × 36 ≈ 62.3538.
Height = 6 × √(2/3) ≈ 4.89898.
Inscribed sphere radius = 6 ÷ √24 ≈ 1.22474.
Circumscribed sphere radius = 6√6 ÷ 4 ≈ 3.67423.

Each point is the calculator re-run with only “Edge length a (regular)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Edge length a (regular)	Volume
1.5	0.397748
3	3.18198
4.5	10.7392
6	25.4558
7.5	49.7184
9	85.9135
12	203.647

How to read the result

Volume is in the input unit cubed and area in the unit squared. In a regular tetrahedron the circumradius is exactly three times the inradius, and the height is four times the inradius, which are handy checks. For general vertices, a volume close to 0 means the points are almost in one plane.

Common mistakes

Forgetting the division by 6 in the vector formula.
Using the edge length as the height.
Taking the triple product without its absolute value, which can give a negative volume.
Entering four points that lie in one plane.

Keep in mind: Results assume exact coordinates. For 3D meshes and engineering parts, very flat tetrahedra can give large relative errors.

Frequently asked questions

What is the volume of a regular tetrahedron with edge 1?

1 ÷ (6√2) ≈ 0.11785.

Why divide the triple product by 6?

A tetrahedron is one sixth of the parallelepiped built on the same three edges.

How many faces, edges and vertices does a tetrahedron have?

4 faces, 6 edges and 4 vertices.

Sources: OpenStax, Calculus Volume 3 (2016), §2.4 The Cross Product (triple scalar product and parallelepiped volume)

Checked 9 Nov 2026. Lumen Learning, Calculus III, “Using the Cross Product” (adapted from OpenStax Calculus Volume 3, Strang & Herman): the volume of a parallelepiped with adjacent edges u, v, w is |u · (v × w)|. The tetrahedron as one sixth of that parallelepiped and the regular-tetrahedron formulas are standard results.

Your numbers