A parallelepiped is a three-dimensional solid bounded by six parallelograms. It can be viewed as the spatial analogue of a parallelogram in the plane: each of its three pairs of opposite faces are parallel, and opposite edges are equal and parallel. In elementary geometry this family of solids includes familiar shapes such as cubes and rectangular boxes as special, orthogonal instances. For a succinct concept link see parallelepiped overview.

Definitions and basic properties

There are several equivalent ways to describe a parallelepiped:

  • As a hexahedron with six faces, each a parallelogram (a polyhedron variant) — see parallelogram faces.
  • As a hexahedron whose faces form three distinct parallel pairs — sometimes called a rhomboid in older texts; compare with rhombohedron.
  • As a prism whose base is a parallelogram, extended along a vector not necessarily perpendicular to the base — see prism.

From these descriptions it follows that opposite faces are congruent parallelograms, there are twelve edges grouped into three parallel directions, and eight vertices where three edges meet. A parallelepiped is a special case of a hexahedron and of a more general affine transform of a cuboid; its rectangular special case is often called a cuboid or rectangular box, while the fully regular special case is the cube — see cuboid and cube.

Geometry and algebraic representation

Algebraically a parallelepiped can be generated from three vectors a, b and c that meet at a single vertex: the twelve edges are translated copies of these generating vectors. The solid consists of all points of the form x = p + s a + t b + u c with 0 ≤ s,t,u ≤ 1 for a chosen origin vertex p. This viewpoint makes it natural to compute volume: the volume equals the absolute value of the scalar triple product of the edge vectors, |a · (b × c)|. That formula emphasizes how skewed edges and non‑orthogonality affect the occupied space.

Faces and diagonals have predictable relationships: each face is a parallelogram spanned by two of the generating vectors, face normals come from cross products of those vectors, and the main body diagonal runs from a vertex p to p + a + b + c. In Euclidean geometry one may additionally discuss angles and orthogonality; in an affine treatment only parallelism and ratios along lines are preserved — see Euclidean vs affine geometry.

Special cases and examples

Important special types of parallelepiped include:

  • The rectangular parallelepiped (all faces rectangles), commonly called a box or cuboid — compare rectangular cuboid.
  • The cube (all faces squares), the regular instance with maximal symmetry — see square and cube.
  • The rhombohedron (all faces are congruent rhombi), a fully skewed but equilateral-faced case — related to rhombus.

Applications and examples appear throughout mathematics and the physical sciences: unit cells in crystal lattices are parallelepipeds when lattice vectors are non‑orthogonal; coordinate transformations and linear algebra use parallelepiped volumes to measure determinants; and in computational geometry they serve for collision bounds and bounding volumes. For related lattice and crystal concepts see rectangles and lattices and general polyhedra discussions at polyhedron.

Historical notes and distinctions

The name derives by analogy from parallelogram, emphasizing parallel opposite faces. Historically classical geometers studied special instances (rectangular boxes and rhombohedra) long before the modern vector formulation simplified volume and transformation calculations. When discussing prisms and hexahedra, note the distinction: a parallelepiped is a prism with a parallelogram base but is more strictly characterized by its three directions of parallel edges; see hexahedron for the broader family and parallelogram for the planar analogue.

For further foundations and formal properties consult resources on solid geometry and linear algebra where vectors, determinants, and affine maps make the structure and calculations involving parallelepipeds transparent — introductory references include entries on parallelepiped, cube, and general prisms at prism.