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. That is the key that unlocks the volume formula below. Like other solids, it belongs to the broader family of 3D geometry shapes.
The Volume Formula: Scalar Triple Product
The most useful fact about a parallelepiped is how to measure the space inside it. Take the three edge vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ that meet at a single corner. The volume is the absolute value of their scalar triple product:
$$V = |\vec{a} \cdot (\vec{b} \times \vec{c})|$$
Each piece has a plain meaning. The cross product $\vec{b} \times \vec{c}$ produces a vector perpendicular to the base parallelogram, and its length equals the base area. Taking the dot product with $\vec{a}$ multiplies that base area by the height of $\vec{a}$ above the base. Base area times height is volume — exactly what the formula computes. For the two operations it relies on, see the cross product of two vectors and the dot product, and for the full derivation of the triple product itself see scalar triple product.
The absolute value matters: the triple product can come out negative depending on the order of the vectors, but volume is never negative, so we take the magnitude. The scalar triple product also equals a determinant, which is the fastest way to compute it by hand:
$$V = \left| \det \begin{bmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{bmatrix} \right|$$
Examples of a Parallelepiped
Each example builds from a direct substitution to a fuller reasoning task. The problem statement is bold; the working is not. One multiplication symbol, $\times$, is used throughout for the cross product.
Example 1
Find the volume of a parallelepiped with edge vectors $\vec{a} = (1, 0, 0)$, $\vec{b} = (0, 2, 0)$, $\vec{c} = (0, 0, 3)$.
These edges are mutually perpendicular, so this is a rectangular box. Using the determinant:
$$V = \left| \det \begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix} \right|$$
$$V = |1 \times (2 \times 3 - 0)| = |6| = 6$$
The volume is 6 cubic units, matching $1 \times 2 \times 3$ as expected for a box.
Example 2
Compute the volume for $\vec{a} = (2, 1, 0)$, $\vec{b} = (1, 3, 0)$, $\vec{c} = (0, 0, 4)$.
First expand the determinant along the third row (which has the single nonzero entry 4):
$$V = \left| \det \begin{bmatrix} 2 & 1 & 0 \ 1 & 3 & 0 \ 0 & 0 & 4 \end{bmatrix} \right|$$
$$V = |4 \times (2 \times 3 - 1 \times 1)|$$
$$V = |4 \times (6 - 1)|$$
$$V = |4 \times 5| = 20$$
The volume is 20 cubic units.
Example 3: The order-of-operations trap
A student computes $V = |\vec{a} \cdot (\vec{b} \times \vec{c})|$ but does the dot product first: they calculate $\vec{a} \cdot \vec{b}$, then try to cross the result with $\vec{c}$.
Following that path immediately breaks down: $\vec{a} \cdot \vec{b}$ is a single number (a scalar), and you cannot take the cross product of a plain number with a vector. The expression $\vec{a} \cdot \vec{b} \times \vec{c}$ has no meaning if the dot is done first.
The rescue is to respect the bracket: the cross product must come first, producing a vector, and only then does the dot product turn it into a scalar. Reading $\vec{a} \cdot (\vec{b} \times \vec{c})$ left to right without the brackets is the error. Always evaluate the inside of the parentheses first.
Example 4
Find the volume for $\vec{a} = (1, 2, 3)$, $\vec{b} = (2, 0, 1)$, $\vec{c} = (0, 1, 2)$.
Expand the determinant along the top row:
$$V = \left| \det \begin{bmatrix} 1 & 2 & 3 \ 2 & 0 & 1 \ 0 & 1 & 2 \end{bmatrix} \right|$$
$$V = |1(0 \times 2 - 1 \times 1) - 2(2 \times 2 - 1 \times 0) + 3(2 \times 1 - 0 \times 0)|$$
$$V = |1(-1) - 2(4) + 3(2)|$$
$$V = |-1 - 8 + 6| = |-3| = 3$$
The volume is 3 cubic units. Note how the raw triple product came out negative before the absolute value.
Example 5
Show that if the three edge vectors are coplanar, the parallelepiped has zero volume.
If $\vec{a}$, $\vec{b}$, $\vec{c}$ all lie in one plane, the solid is flattened — it has no thickness. Take $\vec{a} = (1, 1, 0)$, $\vec{b} = (2, 0, 0)$, $\vec{c} = (0, 3, 0)$, all in the $xy$-plane:
$$V = \left| \det \begin{bmatrix} 1 & 1 & 0 \ 2 & 0 & 0 \ 0 & 3 & 0 \end{bmatrix} \right| = 0$$
The zero column of $z$-values forces the determinant to 0. Zero volume signals coplanar edges — a genuinely flat, degenerate solid.
Example 6
A crystallographer models a mineral cell as a parallelepiped with edges $\vec{a} = (3, 0, 0)$, $\vec{b} = (1, 4, 0)$, $\vec{c} = (0, 0, 5)$ in ångström units. Find its volume.
Expand along the third row:
$$V = |5 \times (3 \times 4 - 0 \times 1)| = |5 \times 12| = 60$$
The unit-cell volume is 60 cubic ångströms. This is exactly how crystal-cell volumes are computed in materials science.
Where the Parallelepiped Earns Its Keep
The parallelepiped is the shape that appears whenever three directions in space do not meet at right angles.
Crystallography. The unit cell of a crystal — the smallest repeating block of atoms — is almost always a parallelepiped, and its volume is found with exactly the scalar-triple-product formula above. Triclinic crystals, where no edge is perpendicular to another, are the general case.
Computer graphics and physics. A "bounding box" that must hug a rotated object is an oriented parallelepiped, and its volume tells an engine how much space to reserve.
Linear algebra. The triple product's link to the determinant means the volume of a parallelepiped is the geometric meaning of a $3 \times 3$ determinant. A determinant of zero says the three vectors are coplanar — the flattened case from Example 5.
The deeper idea is that the determinant is a volume. When Josiah Willard Gibbs formalised vector analysis in the 1880s, this connection between algebra and space became one of its most reused results. For the geometric interpretation in full, see Wolfram's demonstration of the scalar triple product.
The Mistakes Students Make Most Often
Mistake 1: Doing the dot product before the cross product
Where it slips in: Reading $\vec{a} \cdot (\vec{b} \times \vec{c})$ left to right and computing the dot first, as in Example 3.
Don't do this: Evaluate $\vec{a} \cdot \vec{b}$ and then try to cross a scalar with $\vec{c}$ — an undefined operation.
The correct way: Always compute the cross product $\vec{b} \times \vec{c}$ first to get a vector, then dot with $\vec{a}$. The brackets are not decoration; they set the order.
Mistake 2: Forgetting the absolute value
Where it slips in: The triple product returns a negative number and the memoriser reports a negative volume, as nearly happened in Example 4.
Don't do this: State the volume as $-3$ cubic units.
The correct way: Volume is a physical amount of space and can never be negative. Take the magnitude: $|-3| = 3$. The sign only tells you the orientation (handedness) of the vectors, not the size.
Mistake 3: Confusing a parallelepiped with a rectangular box
Where it slips in: Assuming edges are perpendicular and using $V = l \times w \times h$ for a slanted solid.
Don't do this: Multiply three edge lengths together when the edges are not at right angles.
The correct way: The simple length-times-width-times-height rule only works when the edges are mutually perpendicular. For a slanted (oblique) parallelepiped, use the scalar triple product, which automatically accounts for the slant through the perpendicular height.
Conclusion
A parallelepiped is a 3D solid with six parallelogram faces and opposite faces parallel and congruent.
Its volume is the absolute value of the scalar triple product of the three edge vectors, $V = |\vec{a} \cdot (\vec{b} \times \vec{c})|$.
The triple product equals a $3 \times 3$ determinant, giving a quick hand-calculation route.
The most common mistakes are doing the dot before the cross and dropping the absolute value.
Zero volume means the three edges are coplanar, so the solid is degenerate and flat.
A Practical Next Step
Work through the exercises below to solidify your understanding, and verify each with the determinant.
Find the volume of the parallelepiped with edges $(2,0,0)$, $(0,3,0)$, $(0,0,1)$. (Answer to Question 1: 6 cubic units.)
Compute $|\vec{a} \cdot (\vec{b} \times \vec{c})|$ for $\vec{a}=(1,0,1)$, $\vec{b}=(0,1,0)$, $\vec{c}=(1,0,0)$. (Answer to Question 2: 1 cubic unit.)
What does a determinant of 0 tell you about the three edge vectors? (Answer to Question 3: they are coplanar, so the volume is zero.)
To take the parallelepiped further with a teacher, explore Bhanzu's geometry tutor, a high school math tutor for vector methods, or browse math classes online. Want to work through the volume formula with a live trainer? Try a free class.
Read More
Prisms — a related class of solids with matching parallel faces.
Rectangular Prism — the right-angled special case of a parallelepiped.
Multiplication of Vectors — how dot and cross products combine.
Product of Vectors — an overview of the vector products behind the triple product.
Cube Formula — volume of the cube, the simplest parallelepiped.
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