Triangular Pyramid Volume Calculator

A triangular pyramid is a pyramid with a triangular base. With 4 vertices, 6 edges and 4 triangular faces it is also called a tetrahedron, and it is the simplest solid there is: no fewer faces will enclose a volume.

V = (1/3) × A_base × h

Where A_base is the area of the triangular base and h is the perpendicular height from the base plane up to the apex. The base can be any triangle at all, equilateral, isosceles, scalene or right, and the apex does not have to sit over the middle of it. Slide the apex sideways and the volume does not change, exactly as with a 2D triangle.

If the base is a specific kind of triangle, you can substitute the formula for that triangle:

• Right triangle base (legs a, b): V = (1/3) × (½ × a × b) × h = (a × b × h) / 6
• Equilateral triangle base (side s): V = (1/3) × (s² × √3 / 4) × h = (s² × h × √3) / 12

Worked example: how big is a tetrahedral milk carton? The Tetra Classic packs that gave Tetra Pak its name were tetrahedra, formed by sealing a tube of coated paper alternately at right angles. Work backwards from the capacity to find the edge length.

Volume of a regular tetrahedron with side s: V = s³ / (6√2) = s³ × √2 / 12 ≈ 0.1178 × s³.

For a 250 mL pack, 250,000 mm³ = 0.1178 × s³, so s³ = 2,122,000 and s ≈ 128 mm. A 250 mL tetrahedron is about 13 cm along every edge, which is a surprisingly large object to hold.

Run it the other way to see why the cube law bites. Halve the edge to 64 mm and the pack holds 31 mL, an eighth as much, not half. That is the whole reason single-serve tetrahedral packs never scaled below about 100 mL and why the format was eventually squeezed out by the brick.

Where triangular pyramids show up:

• Pyramid-shaped tea bags. Roughly regular tetrahedra with edges around 4 to 5 cm, so 7 to 15 mL of space for the leaves to move in. That interior room, not the mesh area, is the point of the shape.
• Tetrahedral packaging. Beverage cartons of the 1950s and 60s, and some single-serve sauce packets today.
• Architectural elements. Geodesic dome triangles, space frames, some modern art installations.
• Crystallography. The diamond lattice puts every carbon atom at the centre of a tetrahedron of four neighbours.
• Chemistry models. Methane (CH₄) is a regular tetrahedron with carbon at the centre and hydrogens at the vertices.

The regular tetrahedron, the special case:

All four faces equilateral triangles. All edges equal. Highest symmetry of any tetrahedron.

• Side length s.
• Height h = s × √(2/3) ≈ 0.8165 × s.
• Volume V = s³ × √2 / 12 ≈ 0.1178 × s³.
• One of the five Platonic solids.

Counting edges, vertices, faces (Euler check): 4 vertices, 6 edges, 4 faces → V − E + F = 4 − 6 + 4 = 2. ✓ (Euler’s formula for any convex polyhedron.)

Sanity check:

• A_base = 0: V = 0. ✓
• h = 0: V = 0 (flat triangle). ✓
• Regular tetra with s = 1: V = √2 / 12 ≈ 0.1178.