What this calculator does
A parallelepiped is a six-faced solid built from three edge vectors meeting at a common vertex, where opposite faces are parallel but the edges are not necessarily at right angles to each other. A rectangular box is just the special case where all three vectors happen to be perpendicular; a general parallelepiped can be skewed in any direction.
The volume of a parallelepiped formula uses the scalar triple product of its three edge vectors a, b and c: V = |a · (b × c)|. This works whether or not the edges are at right angles, which is exactly why an ordinary box or tank calculator, built on length times width times height, cannot be used for a skewed shape like this.
The formula
First take the cross product of vectors b and c, which gives a new vector at right angles to both. Then take the dot product of vector a with that result, and finally take the absolute value, since the raw scalar triple product can come out negative depending on the order and orientation of the three vectors, but volume itself is never negative.
| Term | Meaning |
|---|---|
| a, b, c | The three edge vectors of the parallelepiped, each starting from the same shared vertex. |
| b × c | The cross product of vectors b and c, a vector perpendicular to both with a length equal to the area of the parallelogram they form. |
| a · (b × c) | The scalar triple product: the dot product of a with that cross product, whose absolute value equals the parallelepiped volume. |
The inputs explained
| Field | What to enter |
|---|---|
| Vector a, x | The x-component of edge vector a. |
| Vector a, y | The y-component of edge vector a. |
| Vector a, z | The z-component of edge vector a. |
| Vector b, x | The x-component of edge vector b. |
| Vector b, y | The y-component of edge vector b. |
| Vector b, z | The z-component of edge vector b. |
| Vector c, x | The x-component of edge vector c. |
| Vector c, y | The y-component of edge vector c. |
| Vector c, z | The z-component of edge vector c. |
When to use it
Working with a crystal unit cell
Crystallography describes a unit cell by three lattice vectors that are often not at right angles; the scalar triple product of those vectors gives the cell volume directly, which feeds into density and packing calculations.
Checking a skewed structural or fabricated shape
A tank, duct or panel that has been built or measured with edges slightly out of square is not a true rectangular box, and using length times width times height on it would overstate or understate its actual volume; the general parallelepiped volume formula handles the true, skewed shape.
Verifying vectors are not coplanar
If the scalar triple product comes out at zero, the three vectors all lie in the same plane and cannot form a genuine three-dimensional solid at all, which is a useful check before relying on them for any three-dimensional calculation.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How volume changes as one edge vector scales up, with the other two fixed
Vector a scaled along the x-axis, with b and c fixed at the values above.
Questions
What is the parallelepiped volume formula?
Volume equals the absolute value of the scalar triple product of the three edge vectors sharing a vertex: V = |a · (b × c)|. This reduces to the familiar length × width × height when the three vectors happen to be mutually perpendicular.
Why take the absolute value?
The scalar triple product a · (b × c) is signed: it comes out negative if the three vectors form a left-handed rather than right-handed set, in the order given. Physical volume is never negative, so the absolute value is taken to get the actual volume regardless of vector order.
What does it mean if the calculated volume is zero?
A volume of zero means the three vectors are coplanar, they all lie flat in the same plane rather than spanning three dimensions. There is no genuine three-dimensional parallelepiped in that case, just a flat parallelogram or a degenerate shape.
How is this different from a normal box volume calculator?
A standard box, tank or cuboid calculator assumes all three edges meet at right angles, so it only ever needs length, width and height as plain numbers. A parallelepiped's edges can point in any direction, so it needs full 3D vectors and the scalar triple product rather than a simple multiplication.
For a right-angled box, tank or container instead, see the volume in litres calculator. For another vector-based geometric quantity, see the second moment of area calculator.