Tetrahedron Volume Calculator
Compute the volume of a regular tetrahedron from its edge length, or of a general tetrahedron using base area and height or the 3D coordinates of its vertices. The tool also returns height, face area, total surface area and sphere radii for the regular case.
Interactive tetrahedron volume calculator
Use any length unit (cm, m, in, ft, …). All outputs will be expressed consistently in that unit.
Results (regular tetrahedron)
- Volume \(V\)
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- Height \(h\)
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- Face area \(A_{\text{face}}\)
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- Total surface area \(A_{\text{tot}}\)
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- Insphere radius \(r\)
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- Circumsphere radius \(R\)
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Enter an edge length and click “Calculate” to see volume and key geometric properties.
Use consistent units (e.g. base in m² and height in m).
Result
Valid for any tetrahedron if \(A\) is the area of a face and \(h\) the perpendicular distance from the opposite vertex.
Enter the coordinates of the four vertices \(A, B, C, D\) in 3D. The volume is computed from the triple scalar product: \( V = \dfrac{|\det(\overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{AD})|}{6} \).
Result
Coordinates are interpreted in arbitrary length units (e.g. meters). The volume is expressed in the corresponding cubic unit.
Volume of a regular tetrahedron
A regular tetrahedron has four congruent equilateral triangular faces and six equal edges of length \(a\). Its volume is smaller than that of a cube with the same edge, but it has highly symmetric geometry that makes the formulas compact and useful in many problems.
Volume formula (regular tetrahedron)
\[ V = \frac{a^3}{6\sqrt{2}} \]
Here \(a\) is the common edge length. If \(a\) is measured in meters, \(V\) is in cubic meters; if \(a\) is in centimeters, \(V\) is in cubic centimeters, and so on.
Other useful regular tetrahedron formulas
- Height: \[ h = \frac{\sqrt{6}}{3} a \]
- Area of each face (equilateral triangle): \[ A_{\text{face}} = \frac{\sqrt{3}}{4} a^2 \]
- Total surface area: \[ A_{\text{tot}} = 4 A_{\text{face}} = \sqrt{3}\, a^2 \]
- Insphere radius: \[ r = \frac{\sqrt{6}}{12} a \]
- Circumsphere radius: \[ R = \frac{\sqrt{6}}{4} a \]
General tetrahedron: volume from base area and height
For any tetrahedron (not necessarily regular), if you know the area \(A\) of one face and the perpendicular height \(h\) from the opposite vertex to that face, the volume is:
General formula with base and height
\[ V = \frac{1}{3} A h \]
This is the natural 3D analogue of the formula “area of a triangle = base × height / 2” and “volume of a pyramid = base area × height / 3”.
General tetrahedron: volume from coordinates
If you know the coordinates of the four vertices \(A, B, C, D\) of a tetrahedron in 3D space, you can compute the volume using vector algebra.
Let \(\overrightarrow{AB} = B - A\), \(\overrightarrow{AC} = C - A\), \(\overrightarrow{AD} = D - A\). Define the triple scalar product:
Triple scalar product and volume
\[ [\overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{AD}] = \det\big(\overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{AD}\big) \] \[ V = \frac{\big|[\overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{AD}]\big|}{6} \]
Geometrically, \(\big|[\overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{AD}]\big|\) is six times the volume of the tetrahedron formed by \(A, B, C, D\). The sign of the determinant depends on the orientation of the three vectors, but the volume is always taken as a non-negative quantity.
FAQ: working with tetrahedron volume
When should I use the regular tetrahedron formula \(V = a^3 / (6\sqrt{2})\)?
Use this formula only when all edges of the tetrahedron have the same length. In that case you get volume, height, face area and surface area with very compact expressions. If the faces are not equilateral or edges differ, use the base & height or coordinate methods instead.
What units does the calculator use for volume?
The calculator keeps units consistent. If you enter edge length in centimeters, volume is in cubic centimeters; if you enter coordinates in meters, volume is in cubic meters. The tool does not convert between different unit systems internally, it simply propagates the unit you choose.
Why do I get zero volume when I enter coordinates?
If the four points are coplanar (lie on the same plane) or if two or more vertices coincide, the tetrahedron degenerates and its volume is zero. Numerically, rounding errors can also produce very small values that are effectively zero for practical purposes.