Triangular Pyramid Volume Calculator

A triangular pyramid is a pyramid with a triangular base. With 4 vertices, 6 edges, and 4 triangular faces, it’s also called a tetrahedron — the simplest of all 3D solids.

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 to the apex. The base triangle can be any shape — equilateral, isosceles, scalene, right.

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 — Tetra Pak pyramid milk packs: The original 1952 Tetra Classic milk packs were tetrahedral, with all four faces being equilateral triangles. Side length ≈ 78 mm (for a 250 mL pack). Volume of a regular tetrahedron with side s: V = s³ / (6√2) = s³ × √2 / 12 ≈ s³ × 0.118. For s = 78 mm: V ≈ 78³ × 0.118 ≈ 56,000 mm³ ≈ 56 mL.

Wait — the marketed capacity was 250 mL, but the calculated geometric volume is only 56 mL? Mismatch.

In reality, Tetra Classic packs of 250 mL were closer to 130 mm on a side, not 78 mm. Bigger pyramid = more volume. For s = 130 mm: V ≈ 130³ × 0.118 ≈ 259,000 mm³ = 259 mL. That checks out — close to the labeled 250 mL with a tiny amount of headroom.

Where triangular pyramids show up:

• Pyramid-shaped tea bags. Tetley and Lipton “pyramid” tea bags are roughly regular tetrahedra. Larger surface area for the leaves to infuse than flat bags. The volume varies but typically ~5-10 mL.
• Tetrahedral packaging. Beverage cartons (the classic 1950s-60s Tetra Pak design), some single-serve packets.
• Architectural elements. Geodesic dome triangles, some modern art installations.
• Crystallography. Diamond crystal lattice — carbon atoms in a tetrahedral configuration.
• Chemistry models. Methane (CH₄) is a regular tetrahedron with carbon at center, hydrogens at vertices.

The “regular tetrahedron” — 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.