Tetrahedron Volume Calculator

A tetrahedron is the simplest three-dimensional solid: a polyhedron with four triangular faces, six edges, and four vertices. It belongs to the family of pyramids — specifically, it is the pyramid whose base is a triangle rather than a square or other polygon. Among all convex polyhedra, the tetrahedron has the fewest faces (four), which gives it exceptional rigidity and makes it a recurring shape in nature and engineering. A regular tetrahedron is one in which all four faces are equilateral triangles of equal size. Because all edges are equal, a regular tetrahedron is completely described by a single measurement: its edge length a. The volume formula is V = a³√2 / 12. This can also be written as V = a³ / (6√2). For example, a regular tetrahedron with edge length 6 has a volume of 216√2 / 12 = 18√2 ≈ 25.456 cubic units. For an irregular tetrahedron — one where the four faces are not all congruent equilateral triangles — the edge-length formula no longer applies. Instead, you can use the base-area-and-height formula that applies to any pyramid: V = (1/3) × A × h, where A is the area of the triangular base and h is the perpendicular height from the base to the opposite vertex (the apex). This formula works regardless of the shape of the base triangle or the angle of the apex. The (1/3) factor in the pyramid formula comes from calculus: if you integrate the cross-sectional area of a pyramid from base to apex, you get one-third of the product of base area and height. This contrasts with a prism, which has a constant cross-section and therefore a volume of A × h with no factor of one-third. Tetrahedra appear throughout science and engineering. In chemistry, the carbon atom in methane (CH₄) and in diamond sits at the center of a tetrahedron whose vertices are occupied by hydrogen atoms or other carbon atoms. This tetrahedral geometry minimises the repulsion between electron pairs around the central atom, following the VSEPR model. In structural engineering, the tetrahedron is the most rigid of all 3-D frames: it is the only polyhedron in which every face is a triangle, and adding a brace to any face creates no additional rigidity. This property drives the design of geodesic domes and space trusses. In computer graphics, complex 3-D surfaces are subdivided into tetrahedral meshes for finite-element analysis and physics simulation.