Parallelepiped: Definition, Volume & Scalar Triple Product

What Is a Parallelepiped?

A parallelepiped is a solid figure whose six faces are all parallelograms, arranged so that opposite faces are parallel and identical. It is the 3D analogue of a parallelogram: just as a parallelogram is a "slanted rectangle," a parallelepiped is a "slanted box." A cube, a rectangular box ( cuboid), and a rhombohedron are all special cases where the faces become squares, rectangles, or rhombi.

A parallelepiped has 8 vertices, 12 edges, and 6 faces. The three edges meeting at any one vertex fully determine the solid — give those three edges as vectors and the whole shape is fixed.

The Volume Formula: Scalar Triple Product

The volume is the absolute value of their scalar triple product:

V=∣a⃗⋅(b⃗×c⃗)∣

The cross product b⃗×c⃗ produces a vector perpendicular to the base parallelogram, and its length equals the base area. Taking the dot product with a⃗ multiplies that base area by the height of a⃗ above the base. Base area times height is volume — exactly what the formula computes.

The absolute value matters: the triple product can come out negative, but volume is never negative, so we take the magnitude. The scalar triple product also equals a determinant:

V=∣det⁡[\begin{bmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{bmatrix}]∣

Examples of a Parallelepiped

Example 1

Find the volume of a parallelepiped with edge vectors a⃗=(1,0,0), b⃗=(0,2,0), c⃗=(0,0,3).

These edges are mutually perpendicular, so this is a rectangular box. Using the determinant:

V=∣det⁡[\begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix}]∣

V=∣1×(2×3−0)∣=∣6∣=6

The volume is 6 cubic units.

Example 2

Compute the volume for a⃗=(2,1,0), b⃗=(1,3,0), c⃗=(0,0,4).

First expand the determinant along the third row:

V=∣det⁡[\begin{bmatrix} 2 & 1 & 0 \ 1 & 3 & 0 \ 0 & 0 & 4 \end{bmatrix}]∣

V=∣4×(2×3−1×1)∣=∣20∣=20

The volume is 20 cubic units.

Example 3: The order-of-operations trap

A student computes V=∣a⃗⋅(b⃗×c⃗)∣ but does the dot product first.

The correct way is to respect the bracket: the cross product must come first.

Example 4

Find the volume for a⃗=(1,2,3), b⃗=(2,0,1), c⃗=(0,1,2).

V=∣det⁡[\begin{bmatrix} 1 & 2 & 3 \ 2 & 0 & 1 \ 0 & 1 & 2 \end{bmatrix}]∣

V=∣−3∣=3

The volume is 3 cubic units.

Example 5

Show that if the three edge vectors are coplanar, the parallelepiped has zero volume.

V=∣det⁡[\begin{bmatrix} 1 & 1 & 0 \ 2 & 0 & 0 \ 0 & 3 & 0 \end{bmatrix}]∣=0

Example 6

A crystallographer models a mineral cell as a parallelepiped with edges a⃗=(3,0,0), b⃗=(1,4,0), c⃗=(0,0,5). Find its volume.

V=∣5×(3×4−0×1)∣=60

The unit-cell volume is 60 cubic ångströms.

Where the Parallelepiped Earns Its Keep

Crystallography: The unit cell of a crystal is almost always a parallelepiped.
Computer graphics and physics: A "bounding box" that must hug a rotated object is an oriented parallelepiped.
Linear algebra: The triple product's link to the determinant means the volume of a parallelepiped is the geometric meaning of a 3×3 determinant.

The Mistakes Students Make Most Often

Mistake 1: Doing the dot product before the cross product

Where it slips in: Computing the dot first.

Mistake 2: Forgetting the absolute value

Where it slips in: The triple product returns a negative number.

Mistake 3: Confusing a parallelepiped with a rectangular box

Where it slips in: Assuming edges are perpendicular.

Conclusion

A parallelepiped is a 3D solid.
Its volume is the absolute value of the scalar triple product of the three edge vectors.
The most common mistakes are doing the dot before the cross and dropping the absolute value.

A Practical Next Step

Work through the exercises below to solidify your understanding:

Find the volume of the parallelepiped with edges (2,0,0), (0,3,0), (0,0,1). (Answer: 6 cubic units.)
Compute ∣a⃗⋅(b⃗×c⃗)∣ for a⃗=(1,0,1), b⃗=(0,1,0), c⃗=(1,0,0). (Answer: 1 cubic unit.)
What does a determinant of 0 tell you about the three edge vectors? (Answer: they are coplanar, so the volume is zero.)