Regular Tetrahedron Volume Calculator
Compute regular tetrahedron volume from a single edge length.
For 4-sided dice (d4), pyramid tea bags, and crystal structure modeling.
A regular tetrahedron is one of the five Platonic solids. All four faces are equilateral triangles, all six edges are equal, and all four vertices are equivalent.
V = s³ / (6√2) = s³ × √2 / 12 ≈ 0.1178 × s³
Where s is the edge length (the same for all six edges).
Worked example: a d4 gaming die A standard 16 mm tabletop d4 has edge length s = 16 mm. V = 16³ × 0.11785 = 482.7 mm³, near enough 0.48 cm³.
At plastic density 1.2 g/cm³ that is 0.58 g per die, so a bag of ten weighs about 6 g. It is the lightest die in the set by a wide margin: a d20 of the same edge holds 8.94 cm³, nearly nineteen times as much material.
Worked example: the Tetra Classic milk pack The 1950s Tetra Pak carton was a regular tetrahedron with s = 130 mm. V = 130³ × 0.1178 ≈ 259,000 mm³ = 259 mL, a comfortable fit for a 250 mL fill with a little headspace.
Tetra Pak chose the shape in 1952 for the machinery, not the material. Run a continuous paper tube through a filler and seal it flat, turning the seal 90° each time, and a string of tetrahedra falls out the end with no forming and almost no offcut. The catch is that tetrahedra do not stack, which is why the brick replaced it, and the tetrahedron is in fact the WORST of the simple solids for surface area per unit volume.
Where regular tetrahedra appear in real measurements:
- d4 dice. The four-sided die of tabletop games, and the one that hurts to step on.
- Tetrahedral kites. Multi-cell kites built from many small tetrahedra, which Alexander Graham Bell pioneered around 1900 precisely because the structure gets stiffer without getting heavier.
- Molecular geometry. Methane and the silicate tetrahedron both place four atoms at the vertices with one at the centre, giving the 109.47° bond angle.
- Diamond structure. Every carbon bonds to four others at the vertices of a tetrahedron, which is where the hardness comes from.
- Tetrahedral packaging. Tetra Classic milk and juice cartons, 1950s to 1970s.
- Teaching models. Platonic-solid sets for geometry and chemistry classes.
Useful tetrahedron measurements (all derived from s):
| Quantity | Formula | Value for s = 1 |
|---|---|---|
| Edge length | s | 1 |
| Face area (equilateral triangle) | (√3 / 4) × s² | 0.433 |
| Face perimeter | 3s | 3 |
| Height (apex to opposite face) | s × √(2/3) | 0.816 |
| Total surface area | √3 × s² | 1.732 |
| Volume | √2 × s³ / 12 | 0.118 |
| Inscribed sphere radius (insphere) | s / (2√6) | 0.204 |
| Circumscribed sphere radius (circumsphere) | s × √6 / 4 | 0.612 |
Volume vs. cube comparison:
A tetrahedron with edge s has volume ~0.118s³. A cube with edge s has volume s³. The tetrahedron holds only 11.8% of the cube’s volume for the same edge length.
For the SAME bounding sphere the gap narrows but stays large. A tetrahedron inscribed in a sphere of radius r has edge 4r/√6, giving V = 8r³/(9√3) ≈ 0.513 r³. A cube in that same sphere has edge 2r/√3 and volume 8r³/(3√3) ≈ 1.540 r³. The cube holds exactly three times as much, since the two expressions differ only by that factor of 3.
Sanity check:
- s = 0: V = 0. ✓
- s = 1: V = √2 / 12 ≈ 0.118. ✓
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This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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