Measure 3D volume from three vectors instantly. Review determinants, signed results, and tetrahedron equivalence clearly. Export clean reports for study, design, validation, and reference.
Enter three 3D vectors. The calculator finds the scalar triple product, signed volume, absolute volume, tetrahedron volume, base area, and height.
This chart compares signed volume, absolute volume, and equivalent tetrahedron volume.
| Case | Vector A | Vector B | Vector C | Signed Volume | Absolute Volume | Tetrahedron Volume | Orientation |
|---|---|---|---|---|---|---|---|
| Unit Cube Basis | (1, 0, 0) | (0, 1, 0) | (0, 0, 1) | 1 | 1 | 0.1667 | Right-Handed |
| Oblique Example | (2, 1, 0) | (1, 0, 2) | (0, 3, 1) | -13 | 13 | 2.1667 | Left-Handed |
| Coplanar Vectors | (1, 2, 3) | (2, 4, 6) | (0, 1, 1) | 0 | 0 | 0 | Zero Volume |
The calculator first builds the cross product of B and C. It then dots vector A with that cross product. The sign shows orientation, while the magnitude gives geometric volume. If the result is zero, the three vectors are coplanar and do not enclose a 3D volume.
It is the 3D volume enclosed by three vectors placed from a common origin. The scalar triple product measures that volume directly.
The sign shows orientation. Positive values usually indicate a right-handed ordering, while negative values indicate a left-handed ordering of the vectors.
Signed volume keeps direction information. Absolute volume removes the sign and shows only the physical size of the parallelepiped.
Zero appears when the vectors are coplanar or linearly dependent. In that case, they do not span a real 3D solid.
A tetrahedron formed by the same three vectors occupies one-sixth of the corresponding parallelepiped, so the conversion divides by six.
The core formula is A · (B × C). It can also be written as the determinant of a 3 × 3 matrix built from the vector components.
Yes. The calculator accepts decimals, negative values, and mixed signs. Those are common in analytic geometry, physics, and engineering work.
If your vector lengths are entered in meters, the area is in square meters and the volume is in cubic meters. The same pattern applies to other units.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.