In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. Three equivalent definitions of parallelepiped are
a hexahedron with three pairs of parallel faces, a polyhedron with six faces (hexahedron), each of which is a parallelogram, and a prism of which the base is a parallelogram. The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all special cases of parallelepiped. Parallelepiped is now usually pronounced or ; traditionally, it was PARR-ə-lel-EP-ih-ped due to its etymology in Ancient Greek παραλληλεπίπεδον (parallēlepípedon) (with a short -i-), meaning a body "having parallel planes". Parallelepipeds are a subclass of the prismatoids.
Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism. A parallelepiped has three sets of four parallel edges; the edges within each set are of equal length. Parallelepipeds result from linear transformations of a cube (for the non-degenerate cases: the bijective linear transformations). Since each face has point symmetry, a parallelepiped is a zonohedron. Also the whole parallelepiped has point symmetry Ci (see also triclinic). Each face is, seen from the outside, the mirror image of the opposite face. The faces are in general chiral, but the parallelepiped is not. A space-filling tessellation is possible with congruent copies of any parallelepiped.
Volume
A parallelepiped is a prism with a parallelogram as base. Hence the volume V {\displaystyle V} of a parallelepiped is the product of the base area B {\displaystyle B} and the height h {\displaystyle h} (see diagram). With
B = | a | ⋅ | b | ⋅ sin γ = | a × b | {\displaystyle B=\left|\mathbf {a} \right|\cdot \left|\mathbf {b} \right|\cdot \sin \gamma =\left|\mathbf {a} \times \mathbf {b} \right|} (where γ {\displaystyle \gamma } is the angle between vectors a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } ), and
h = | c | ⋅ | cos θ | {\displaystyle h=\left|\mathbf {c} \right|\cdot \left|\cos \theta \right|} (where θ {\displaystyle \theta } is the angle between vector c {\displaystyle \mathbf {c} } and the normal to the base), one gets:
V = B ⋅ h = ( | a | | b | sin γ ) ⋅ | c | | cos θ | = | a × b | | c | | cos θ | = | ( a × b ) ⋅ c | . {\displaystyle V=B\cdot h=\left(\left|\mathbf {a} \right|\left|\mathbf {b} \right|\sin \gamma \right)\cdot \left|\mathbf {c} \right|\left|\cos \theta \right|=\left|\mathbf {a} \times \mathbf {b} \right|\left|\mathbf {c} \right|\left|\cos \theta \right|=\left|\left(\mathbf {a} \times \mathbf {b} \right)\cdot \mathbf {c} \right|.}
The mixed product of three vectors is called triple product. It can be described by a determinant. Hence for a = ( a 1 , a 2 , a 3 ) T , b = ( b 1 , b 2 , b 3 ) T , c = ( c 1 , c 2 , c 3 ) T , {\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3})^{\mathsf {T}},~\mathbf {b} =(b_{1},b_{2},b_{3})^{\mathsf {T}},~\mathbf {c} =(c_{1},c_{2},c_{3})^{\mathsf {T}},} the volume is:
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