Tetrahedron Volume Calculator

A tetrahedron is the simplest 3D shape you can build from flat sides: four triangles, four corners, six edges. When every edge has the same length, you get a regular tetrahedron and all four faces become equilateral triangles. That's the case this calculator handles. Type in the edge length and you'll get back the height, volume, and total surface area.

What you're working with

The regular tetrahedron is one of the five Platonic solids. A methane molecule has tetrahedral geometry, with four hydrogen atoms at the corners and a carbon at the centre. The same bond angle of roughly 109.5° is part of why diamond is so hard. Architects use the shape as the building block for space frames because a tetrahedron can't deform without stretching one of its edges. That rigidity is why it keeps showing up in structural design and crystallography.

Using the calculator

Type a number into the edge length field. Height, volume, and surface area fill in right away. Each field has its own unit menu, so you can mix metric and US units freely. Length picks from millimetres, centimetres, metres, inches, or feet; area and volume use the matching squared and cubed versions.

The formulas

Everything keys off the edge length, which we'll call a.

V = \frac{a^3 \sqrt{2}}{12}

A = \sqrt{3}\, a^2

h = \frac{\sqrt{6}}{3}\, a

For a quick sanity check, take a tetrahedron with 10 cm edges. Volume works out to $(1000 \times \sqrt{2}) / 12$ , around $117.85 \space \text{cm}^3$ . Surface area is $\sqrt{3} \times 100$ , or about $173.21 \space \text{cm}^2$ . Height from base to apex is $(\sqrt{6} / 3) \times 10$ , around 8.16 cm. The square roots come from dropping a perpendicular from the apex to the centroid of the base triangle and letting the Pythagorean theorem handle the rest.

Where this comes up

Chemistry uses the tetrahedron for molecular geometry. Carbon's four bonds splay out at the tetrahedral angle of about 109.5°, which is what shapes methane and diamond. Engineers and architects use tetrahedral trusses for the same reason builders use triangles in 2D: a triangle won't squash without one of its sides changing length, and the tetrahedron is the 3D version of that. In computer graphics, mesh volumes are usually computed by chopping a shape into tetrahedra and adding them up. Geometry classes also start here, with the regular polyhedra, before working out to messier solids.

Frequently asked questions

Does this work for irregular tetrahedrons?

Not really. Every formula here assumes all six edges are the same length. For an irregular tetrahedron you'll want the Cayley–Menger determinant, which takes the six edge lengths and gives back the volume.

Why are the formulas full of square roots?

Because the apex isn't directly above any corner of the base; it sits over the centroid, at an angle. When you drop a perpendicular from apex to centroid, the right triangle you carve out has sides involving $\sqrt{2}\text{, }\sqrt{3}\text{, }\text{ and }\sqrt{6}$ . The numbers are irrational, but a few decimal places are plenty for any real-world answer.

Can I get the inscribed or circumscribed sphere radius?

Not from this calculator directly, but the three sphere radii all scale linearly with the edge length. The insphere is $(\sqrt{6} / 12) \cdot a$ , the midsphere is ( $(\sqrt{2} / 4) \cdot a$ , and the circumsphere is $(\sqrt{6} / 4) \cdot a$ . Plug in your edge length and multiply.