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A tetrahedron — the simplest possible 3D solid, with just four triangular faces, four vertices, and six edges — has its volume computed two different ways depending on what you know: a compact shortcut formula for the regular case, where all six edges are exactly equal in length, or the more general scalar-triple-product (determinant) formula, which works for ANY four points in 3D space, regardless of shape or edge lengths. Tetrahedra show up as the fundamental building block in 3D mesh modeling and finite element analysis (any complex 3D shape can be decomposed into tetrahedra for simulation), in chemistry (the tetrahedral geometry of a methane molecule's four hydrogen atoms around a central carbon), and in crystallography and molecular geometry more broadly.
How it works
For a regular tetrahedron, enter just the edge length e; the calculator applies the closed-form shortcut V = e³/(6√2), derived from the general formula applied to a regular tetrahedron's specific symmetric coordinates. For a general (possibly irregular) tetrahedron, enter the (x, y, z) coordinates of all four vertices. The calculator forms three edge vectors, all starting from one chosen vertex (A) and reaching each of the other three (B, C, D): AB, AC, and AD. It then computes their scalar triple product, AB · (AC × AD) — a single number that equals the signed volume of the parallelepiped (a 3D "box" shape) these three vectors would span if extended into a full box. Since a tetrahedron is exactly one-sixth of that parallelepiped (a well-known geometric fact from decomposing a parallelepiped into six equal tetrahedra), the final volume is the absolute value of that scalar triple product divided by 6.
- Enter mode.
- Enter edge length (regular mode).
- Enter vertex A (x, y, z) — general mode.
- Enter vertex B (x, y, z) — general mode.
- Enter vertex C (x, y, z) — general mode.
- Enter vertex D (x, y, z) — general mode.
- Click Calculate to see your results.
Examples
Regular tetrahedron, edge = 4
V = 4³/(6√2) = 64/8.485 ≈ 7.54 cubic units.
General: A=(0,0,0), B=(1,0,0), C=(0,1,0), D=(0,0,1)
Edge vectors AB=(1,0,0), AC=(0,1,0), AD=(0,0,1). Scalar triple product = 1×(1×1−0×0) = 1. Volume = |1|/6 ≈ 0.1667.
General: A=(0,0,0), B=(2,0,0), C=(0,3,0), D=(0,0,4)
This is a right-angle "corner" tetrahedron with mutually perpendicular legs of length 2, 3, and 4. Edge vectors AB=(2,0,0), AC=(0,3,0), AD=(0,0,4). AC × AD = (12,0,0), so AB · (12,0,0) = 24. Volume = |24|/6 = 4 — matching the simpler shortcut for this special case, V = (1/6)×a×b×c = (1/6)×2×3×4 = 4.
Who should use it
- Computing the volume of a tetrahedral mesh element in 3D modeling or finite element analysis.
- Geometry and linear algebra coursework on the scalar triple product.
- Chemistry problems involving tetrahedral molecular geometry.
Industry applications
- 3D modeling and finite element analysis
- Chemistry and crystallography
- Linear algebra education
Advantages
- Handles both the simple regular case and any general, irregular tetrahedron.
- The general mode works for genuinely arbitrary 3D vertex positions.
Limitations
- The regular-tetrahedron shortcut is only valid when all edges are truly equal — otherwise the general vertex-based method must be used instead.
Common mistakes to avoid
- Entering vertex coordinates in the wrong order (should be "x, y, z", comma-separated) for the general mode.
- Using the regular-tetrahedron shortcut formula on a shape that isn't actually regular (edges not all equal), which produces a meaningless result.
- Forgetting to take the absolute value of the scalar triple product before dividing by 6, since the raw signed value can come out negative.
Best practices
- Confirm all six edges are genuinely equal before using the regular-tetrahedron shortcut formula.
- For the general mode, double check each vertex's coordinates are entered as three comma-separated numbers in x, y, z order.
- If the scalar triple product comes out very close to zero, check whether your four points might actually be coplanar (not forming a true tetrahedron).
Tips
- Working with a solid that has a rectangular or triangular base instead? See the Right Rectangular Pyramid Calculator or Triangular Prism Calculator.