What is the formula for the volume of a regular tetrahedron?

V = (a³ times sqrt(2)) / 12, where a is the length of any edge. Because all four faces of a regular tetrahedron are equilateral triangles and all six edges are equal, the single edge length a fully determines the volume. The sqrt(2) factor comes from the geometry of the equilateral triangle base and the perpendicular height.

Why does the formula contain sqrt(2)?

The perpendicular height of a regular tetrahedron with edge a is h = a times sqrt(2/3). The base is an equilateral triangle with area (sqrt(3) / 4) times a². Multiplying (1/3) times base area times height and simplifying gives (a³ times sqrt(2)) / 12. The sqrt(2) enters through the combination of the sqrt(3) in the base area and the sqrt(2/3) in the height.

How do I find the edge length if I know the volume?

Rearrange the formula: a = (12V / sqrt(2))^(1/3). Switch the Solve for dropdown to Edge length, enter the known volume, and the calculator applies this cube root automatically with full working shown.

Does this calculator work for irregular tetrahedra?

No. This calculator implements the formula for a regular tetrahedron (all six edges equal). For an irregular tetrahedron with four arbitrary vertices, the volume is found using the scalar triple product of three edge vectors from one vertex: V = (1/6) times the absolute value of (a dot (b cross c)), where a, b, c are the vectors to the three adjacent vertices.

What units can I use?

You can enter values in millimetres, centimetres, metres, inches, or feet. For volume mode the result is in the matching cubic unit (mm³, cm³, m³, in³, or ft³). For edge mode the result is in the same linear unit as your input. To convert the result to a different unit afterwards, use a unit-conversion tool.

Tetrahedron Volume Calculator

Q: What is a regular tetrahedron?

A regular tetrahedron is a Platonic solid with four faces, each an equilateral triangle, four vertices, and six edges of equal length. It is the simplest convex polyhedron and the three-dimensional analogue of an equilateral triangle. The formula in this calculator applies only to the regular form where all edges are equal.

A tetrahedron volume calculator for the regular tetrahedron — one of the five Platonic solids — that works in both directions: supply the edge length to get the volume, or supply the volume to recover the edge length. Whether you need the calculation for a geometry assignment, a 3-D printing project, or an engineering design that uses tetrahedral voids, the step-by-step working lets you follow and verify every arithmetic move.

What is a regular tetrahedron?

A regular tetrahedron is a solid with four faces, each an equilateral triangle, four vertices, and six edges all of the same length. It is the simplest Platonic solid and appears throughout nature and engineering — from the tetrahedral bonding of carbon atoms to the structural frames used in space-frame roofs. Because every edge is equal, the entire geometry is determined by a single number: the edge length a.

The formula

For a regular tetrahedron with edge length a:

V = (a³ × sqrt(2)) / 12

The derivation proceeds in two steps. First, the base is an equilateral triangle with side a and area (sqrt(3) / 4) * a². Second, the perpendicular height from the base to the opposite vertex is h = a * sqrt(2/3). Substituting into the general pyramid formula V = (1/3) * base area * height gives:

V = (1/3) * (sqrt(3)/4) * a^2 * a * sqrt(2/3)
  = (a^3 / 12) * sqrt(3) * sqrt(2/3)
  = (a^3 / 12) * sqrt(2)

To recover the edge length from a known volume, rearrange by multiplying both sides by 12, dividing by sqrt(2), and taking the cube root:

a = (12V / sqrt(2))^(1/3)

Both paths are computed and displayed with full working in the tool above.

Worked example

A regular tetrahedron has an edge length of 5 cm. What is its volume?

Cube the edge: 5³ = 125
Multiply by sqrt(2): 125 × 1.414214 = 176.777
Divide by 12: 176.777 / 12 = 14.731 cm³

Now reverse the question: a tetrahedral cavity in a metal part must hold 29.463 cm³. What edge length is needed?

Multiply the volume by 12: 29.463 × 12 = 353.556
Divide by sqrt(2): 353.556 / 1.414214 = 250.000
Take the cube root: 250^(1/3) = 6.30 cm

Both calculations run instantly in the tool, with each intermediate step printed on screen.

Relationship to other Platonic solids

The regular tetrahedron is the most compact of the five Platonic solids for a given edge length in the sense that it has the smallest volume-to-surface-area ratio. For comparison, if all five Platonic solids share the same edge length a:

Solid	Volume formula
Tetrahedron (4 faces)	(a³ × sqrt(2)) / 12
Cube (6 faces)	a³
Octahedron (8 faces)	(a³ × sqrt(2)) / 3
Dodecahedron (12 faces)	(a³ / 4) × (15 + 7 × sqrt(5))
Icosahedron (20 faces)	(5/12) × a³ × (3 + sqrt(5))

The tetrahedron row is the one computed here. The octahedron’s formula is exactly four times larger — two regular tetrahedra and an octahedron together tile three-dimensional space, a fact exploited in octet-truss structures.

Practical uses

3-D printing and CNC: verifying that a tetrahedral void or boss will fit a target material volume or weight before slicing.
Chemistry and materials science: computing unit-cell volumes for tetrahedral coordination structures (diamond lattice, silicates, semiconductors).
Geometry homework: showing step-by-step working for volume and reverse-solve problems.
Architecture and structural engineering: sizing tetrahedral nodes in space-frame and truss roof systems.
Gaming and 3-D assets: checking polyhedral die or decorative piece volumes for resin-casting calculations.

Keep the input in a consistent unit and the result will be in the matching cubic unit automatically.