Tetrahedron Volume Calculator
A regular tetrahedron has four identical equilateral-triangle faces and six equal edges. Enter the edge length (or reverse-solve from the volume) to get every geometric property in one click: volume, surface area, height, all three sphere radii, and the surface-to-volume ratio.
Formula
Worked example
For edge length a = 6 cm: volume = 6³ ÷ (6√2) = 216 ÷ 8.485 ≈ 25.4558 cm³. Surface area = √3 × 36 ≈ 62.354 cm², each face (√3/4) × 36 ≈ 15.588 cm². Height = 6 × √(2/3) ≈ 4.899 cm. Inradius = 6 ÷ (2√6) ≈ 1.225 cm. Midsphere = 6 ÷ (2√2) ≈ 2.121 cm. Circumradius = (6√6)/4 ≈ 3.674 cm. SVR = 6√6 ÷ 6 ≈ 2.449 cm^-1. Reverse-solve check: cbrt(25.4558 × 6√2) = cbrt(216) = 6 cm.
How the tetrahedron volume formula works
A regular tetrahedron is one of the five Platonic solids: a pyramid built from four identical equilateral triangles, with three faces meeting at each of its four vertices. Because every edge has the same length a, a single measurement fixes the entire shape. The volume follows the general pyramid rule, one third of the base area times the height: the base is an equilateral triangle of area (√3/4)·a², and the apex sits a height a·√(2/3) above it. Multiplying these and simplifying gives the compact result V = a³ / (6√2), approximately 0.1178·a³. The edge length is cubed, so the volume is extremely sensitive to size: a tetrahedron with twice the edge length holds eight times as much volume. To reverse the calculation, if you know the volume V, the edge length is a = cbrt(V × 6√2), which this calculator handles automatically in reverse-solve mode.
Surface area, height, inradius, midsphere, and circumradius
The total surface area is four equilateral faces: A = √3·a², approximately 1.732·a². The height from any vertex to the centre of the opposite face is h = a·√(2/3), about 0.8165·a. Three concentric spheres share the same centre as the tetrahedron. The inscribed sphere (inradius r_i = a / (2√6) ≈ 0.2041·a) is the largest sphere that fits inside touching all four faces. The midsphere (r_k = a / (2√2) ≈ 0.3536·a) is tangent to all six edges, sitting halfway between the other two in a precise ratio. The circumscribed sphere (circumradius R = a·√6 / 4 ≈ 0.6124·a) passes through all four vertices. For a regular tetrahedron, the circumradius is always exactly three times the inradius, a fixed relationship unique among the Platonic solids. The surface-to-volume ratio, S/V = 6√6 / a, is a useful design metric: a smaller value means more interior space relative to surface, while a larger value indicates a high-surface compact form.
Reverse-solve: finding the edge from a known volume
In many practical situations you know the target volume rather than the edge length. A packaging designer who needs a tetrahedral container holding exactly 100 cm³ of liquid, or an engineer who needs a structural node of known mass, both need to work backwards. Because V = a³ / (6√2), solving for a gives a = (6√2 · V)^(1/3). This is the cube root of six times the square root of two times the volume. Use the "Solve from volume" mode in this calculator to get the edge and every other property from the volume alone.
The volume-to-surface scaling chart
The interactive chart shows how volume and surface area scale together as the edge length varies. Volume grows as the cube, while surface area grows only as the square, so for larger tetrahedra the ratio of surface to volume decreases. This explains why large crystals, large bubbles, and large structural nodes are more efficient per unit of surface than small ones, a principle that shapes everything from soap foam geometry to structural engineering.
Where tetrahedra show up
The tetrahedron is the most rigid simple polyhedron, which is why its shape appears throughout engineering and nature. Space frames and geodesic structures rely on tetrahedral units because a triangle cannot deform without changing its edge lengths. In chemistry, the carbon atom bonds to four neighbours arranged at the vertices of a tetrahedron, giving the famous 109.47 degree bond angle of methane and diamond. Tetrahedral packaging uses the minimum surface area to enclose a given volume compared to other pyramid forms, and the shape tiles three-dimensional space when combined with octahedra, a packing used in molecular models, architectural trusses, and foam structures. The fill-cost estimator in this calculator lets designers price tetrahedral resin molds, concrete forms, or custom containers directly from the geometry.
Regular tetrahedron properties by edge length
| Edge a | Volume | Surface area | Height | Inradius | Midsphere | Circumradius | SVR |
|---|---|---|---|---|---|---|---|
| 1 | 0.1178 | 1.7321 | 0.8165 | 0.2041 | 0.3536 | 0.6124 | 14.697 |
| 2 | 0.9428 | 6.9282 | 1.6330 | 0.4082 | 0.7071 | 1.2247 | 7.348 |
| 5 | 14.7314 | 43.3013 | 4.0825 | 1.0206 | 1.7678 | 3.0619 | 2.939 |
| 6 | 25.4558 | 62.3538 | 4.8990 | 1.2247 | 2.1213 | 3.6742 | 2.449 |
| 10 | 117.851 | 173.205 | 8.165 | 2.041 | 3.536 | 6.124 | 1.470 |
| 20 | 942.809 | 692.820 | 16.330 | 4.082 | 7.071 | 12.247 | 0.735 |
All values use V = a³/(6√2), A = √3·a², h = a√(2/3), r_i = a/(2√6), r_k = a/(2√2), R = a√6/4.
Frequently asked questions
What is the volume formula for a regular tetrahedron?
For a regular tetrahedron with edge length a, the volume is V = a³ / (6√2), approximately 0.1178 × a³. This comes from the pyramid rule: one third of the equilateral base area (√3/4)·a² times the height a·√(2/3).
How do I find the edge length if I only know the volume?
Rearrange V = a³ / (6√2) to get a = (6√2 × V)^(1/3), the cube root of six times the square root of two times the volume. Use the reverse-solve mode in this calculator: select "Edge length from volume" and enter your volume. All other properties are then computed automatically from the derived edge length.
What is the midsphere of a tetrahedron?
The midsphere (also called the intersphere) is the sphere that is tangent to all six edges of the regular tetrahedron, touching each edge at its midpoint. Its radius is r_k = a / (2√2), approximately 0.3536·a. It sits between the inscribed sphere (inradius ≈ 0.2041·a) and the circumscribed sphere (circumradius ≈ 0.6124·a).
What is the surface-to-volume ratio of a regular tetrahedron?
The surface-to-volume ratio is S/V = √3·a² / (a³/(6√2)) = 6√6 / a. It is inversely proportional to edge length: the bigger the tetrahedron, the smaller the ratio. This is useful in chemistry and packaging where a low ratio means more interior volume per unit of surface area.
Does this calculator work for irregular tetrahedra?
No. This tool assumes a regular tetrahedron, where all six edges are equal and all four faces are identical equilateral triangles. An irregular tetrahedron with different edge lengths requires the Cayley-Menger determinant or vector cross-product methods with vertex coordinates.
How much of a cube does a regular tetrahedron fill?
A regular tetrahedron can be inscribed in a cube so that its six edges are the face diagonals of the cube. In that configuration, the tetrahedron occupies exactly one third of the cube's volume. Relative to the smallest bounding cube, the fill fraction is smaller, around 12%.