Tetrahedron Volume Calculator
A tetrahedron is a polyhedron with four triangular faces, four vertices, and six edges — the simplest possible three-dimensional solid. It is a special case of a triangular pyramid where every face is a triangle. When all four faces are equilateral triangles of the same size, the shape is called a regular tetrahedron, one of the five Platonic solids. Tetrahedra appear throughout structural engineering, molecular chemistry (the tetrahedral bond geometry of carbon), crystallography, and computer graphics.
Four Calculation Modes
This calculator supports four independent ways to compute a tetrahedron's volume, so you can use whichever information you have available.
Mode 1: Regular Tetrahedron (Edge Length)
When all six edges have the same length a, the volume formula simplifies to:
V = a³ / (6√2)
For example, if a = 5 cm then V = 125 / (6 × 1.41421…) ≈ 14.73 cm³. This formula is derived from the scalar triple product of the three edge vectors of a regular tetrahedron placed at the origin, then divided by six.
Mode 2: Base Area + Perpendicular Height
Like any pyramid, a tetrahedron obeys the general pyramid volume formula:
V = (1/3) × A × h
where A is the area of the triangular base and h is the perpendicular height from the base plane to the opposite vertex. Example: base area = 10.83 cm², h = 4.08 cm → V ≈ 14.73 cm³, confirming the result from Mode 1 for a regular tetrahedron with a = 5 cm.
Mode 3: Three Edge Vectors (Scalar Triple Product)
If you have three edge vectors a, b, and c emanating from a common vertex, the volume is:
V = (1/6) |a · (b × c)|
Here, b × c is the cross product of vectors b and c, giving a vector perpendicular to the face opposite vertex a. The dot product with a then measures the parallelepiped volume, and dividing by six gives the tetrahedron volume (a parallelepiped can be divided into exactly six congruent tetrahedra). If the scalar triple product equals zero, the three vectors are coplanar and the tetrahedron has zero volume.
Mode 4: Four 3D Vertices (Coordinates)
When you know the four vertex coordinates — for example from a mesh, CAD model, or geometry problem — enter them as (x, y, z) triples. The calculator internally computes:
- Edge vectors from vertex 1:
v₂ − v₁,v₃ − v₁, andv₄ − v₁ - Scalar triple product of those three vectors
- V = (1/6) × |triple product|
Example: vertices at (0,0,0), (5,0,0), (0,5,0), and (0,0,5) give edge vectors (5,0,0), (0,5,0), (0,0,5), yielding a triple product of 125 and V = 125/6 ≈ 20.83 cm³.
Output Units
Results are displayed simultaneously in six volume units: cm³, m³,mm³, in³, ft³, and litres. Internally, all dimensional inputs are converted to metres before computing the SI volume, which is then converted to each output unit using exact conversion factors (1 m³ = 10⁶ cm³ = 10⁹ mm³ = 61,023.74 in³ = 35.3147 ft³ = 1,000 L).
Understanding the Scalar Triple Product
The scalar triple product a · (b × c) is the signed volume of the parallelepiped spanned by the three vectors. Its sign indicates the handedness of the orientation (left- or right-handed coordinate system), but volume is always positive, so the absolute value is used. This geometric identity is fundamental in linear algebra and is used widely in physics for computing work done by a torque and in electromagnetism.
Real-World Applications
- Molecular chemistry: carbon, silicon, and many other atoms form tetrahedral bonds. Volume calculations support molecular modelling and drug design.
- Finite element analysis: tetrahedral meshes are the standard 3D mesh element in structural, thermal, and fluid simulations.
- 3D printing: computing the material volume of tetrahedral objects to estimate resin or filament consumption.
- Architecture: pyramidal glass-and-steel structures such as the Louvre Pyramid use tetrahedral geometry.
- Education: tetrahedron volume problems are standard exercises in 3D geometry, linear algebra, and calculus courses.
Tips for Accurate Results
In Mode 2, ensure that A is the actual triangular face area and his the perpendicular height from the centroid of that face to the opposite vertex — not a slant or edge length. In vector and vertex modes, verify that the three vectors (or the three computed edge vectors) are not coplanar; if the result is zero, check that at least one coordinate differs in each vector's components.