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Parallelepiped

Parallelepiped

A polyhedron with six faces, all of which are parallelograms.

 

3D geometric solid with six parallelogram faces, shown in oblique view with visible parallel edges and slanted sides.

 

 

See also

Rectangular parallelepiped

Key Formula

V=∣a⃗⋅(b⃗×c⃗)∣V = |\vec{a} \cdot (\vec{b} \times \vec{c})|
Where:
  • VV = Volume of the parallelepiped
  • a⃗\vec{a} = Edge vector along the first direction
  • b⃗\vec{b} = Edge vector along the second direction
  • c⃗\vec{c} = Edge vector along the third direction
  • b⃗×c⃗\vec{b} \times \vec{c} = Cross product of vectors b and c, giving a vector perpendicular to both
  • a⃗⋅(b⃗×c⃗)\vec{a} \cdot (\vec{b} \times \vec{c}) = Scalar triple product of the three edge vectors

Worked Example

Problem: Find the volume of a parallelepiped defined by the edge vectors a = (1, 0, 0), b = (0, 3, 0), and c = (0, 0, 4).
Step 1: Compute the cross product b × c.
b⃗×c⃗=∣i^j^k^030004∣=(12,0,0)\vec{b} \times \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{vmatrix} = (12, 0, 0)
Step 2: Compute the dot product a · (b × c). This is the scalar triple product.
a⃗⋅(b⃗×c⃗)=(1)(12)+(0)(0)+(0)(0)=12\vec{a} \cdot (\vec{b} \times \vec{c}) = (1)(12) + (0)(0) + (0)(0) = 12
Step 3: Take the absolute value to get the volume.
V=∣12∣=12V = |12| = 12
Answer: The volume of the parallelepiped is 12 cubic units. (Note: since the vectors are mutually perpendicular, this is equivalent to a rectangular box with dimensions 1 × 3 × 4.)

Another Example

This example uses non-perpendicular edge vectors (b is slanted relative to a), showing that the scalar triple product formula works for any parallelepiped — not just rectangular boxes.

Problem: Find the volume of a parallelepiped with edge vectors a = (2, 0, 0), b = (1, 3, 0), and c = (0, 1, 5).
Step 1: Compute the cross product b × c using the determinant formula.
b⃗×c⃗=∣i^j^k^130015∣=(15−0)i^−(5−0)j^+(1−0)k^=(15,−5,1)\vec{b} \times \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 3 & 0 \\ 0 & 1 & 5 \end{vmatrix} = (15 - 0)\hat{i} - (5 - 0)\hat{j} + (1 - 0)\hat{k} = (15, -5, 1)
Step 2: Compute the scalar triple product a · (b × c).
a⃗⋅(b⃗×c⃗)=(2)(15)+(0)(−5)+(0)(1)=30\vec{a} \cdot (\vec{b} \times \vec{c}) = (2)(15) + (0)(-5) + (0)(1) = 30
Step 3: Take the absolute value for the volume.
V=∣30∣=30V = |30| = 30
Answer: The volume of the parallelepiped is 30 cubic units.

Frequently Asked Questions

What is the difference between a parallelepiped and a rectangular prism?
A rectangular prism (also called a rectangular parallelepiped or cuboid) is a special case of a parallelepiped where all six faces are rectangles and all edges meet at right angles. A general parallelepiped has faces that are parallelograms, so its edges can be slanted. Every rectangular prism is a parallelepiped, but not every parallelepiped is a rectangular prism.
How do you pronounce parallelepiped?
The standard pronunciation is "par-uh-LEL-uh-pie-ped" (rhyming with "pied"). An older, also accepted pronunciation is "par-uh-lel-EP-ih-ped." Both are widely used in mathematics and physics.
Why does the scalar triple product give the volume of a parallelepiped?
The cross product b × c produces a vector whose magnitude equals the area of the parallelogram base formed by b and c. The dot product of a with this vector then projects a onto the direction perpendicular to the base, giving the height. Multiplying base area by height yields volume, which is exactly what the scalar triple product computes.

Parallelepiped vs. Rectangular Parallelepiped (Cuboid)

ParallelepipedRectangular Parallelepiped (Cuboid)
FacesSix parallelogramsSix rectangles
Angles between edgesCan be any angleAll 90°
Volume formulaV=∣a⃗⋅(b⃗×c⃗)∣V = |\vec{a} \cdot (\vec{b} \times \vec{c})|V=l×w×hV = l \times w \times h
RelationshipGeneral caseSpecial case of a parallelepiped

Why It Matters

Parallelepipeds appear throughout multivariable calculus and linear algebra. The volume formula using the scalar triple product is foundational for understanding determinants — the determinant of a 3×3 matrix equals the signed volume of the parallelepiped formed by its column (or row) vectors. This connection makes parallelepipeds essential for grasping concepts like change of variables in triple integrals and the geometric meaning of linear transformations.

Common Mistakes

Mistake: Forgetting to take the absolute value of the scalar triple product.
Correction: The scalar triple product can be negative depending on the orientation (handedness) of the vectors. Volume is always positive, so you must take the absolute value: V=∣a⃗⋅(b⃗×c⃗)∣V = |\vec{a} \cdot (\vec{b} \times \vec{c})|.
Mistake: Confusing a parallelepiped with a prism.
Correction: A prism has two congruent polygonal bases connected by rectangular faces. A parallelepiped is specifically a prism whose bases are parallelograms and whose lateral faces are also parallelograms. Not all prisms are parallelepipeds (e.g., a triangular prism is not).

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