Difference between revisions of "Parallelepiped"

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A '''parallelepiped''' is a [[prism]] that has [[parallelograms]] for its faces. Similarly, a '''parallelepiped''' is equivalently a [[hexahedron]] with six [[parallelogram]]. Specific '''parallelepipeds''' include the [[cube]], the [[cuboid]], and any rectangular [[prism]].
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A '''parallelepiped''' is a [[prism]] that has [[parallelograms]] for its faces. Similarly, a parallelepiped is equivalently a [[hexahedron]] with six parallelogram faces. Specific parallelepipeds include the [[cube]], the [[cuboid]], and any rectangular [[prism]].
  
 
==Specific Cases==
 
==Specific Cases==
A parallelepiped with all rectangular faces is a [[cuboid]], and a parallelepiped with six rhombus faces is known as a [rhombohedron]]. In an <math>n-</math>dimensional space, a parallelepiped is sometimes referred to as an <math>n-</math>dimensional parallelepiped, or as an <math>n-</math>parallelepiped. A [[cube]] is a parallelepiped with all [[square]] faces.
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A parallelepiped with all rectangular faces is a cuboid, and a parallelepiped with six rhombus faces is known as a [[rhombohedron]]. In an <math>n-</math>dimensional space, a parallelepiped is sometimes referred to as an <math>n-</math>dimensional parallelepiped, or as an <math>n-</math>parallelepiped. A cube is a parallelepiped with all [[square]] faces.
  
 
==Volume==
 
==Volume==
The [[volume]] of a parallelepiped is the product of area of one of its faces times the perpendicular distance to the corresponding top face. Alternately, if the three edges of a parallelepiped that meet at one [[vertex]] are defined as [[vectors]] <math>a, b,</math> and <math>c</math> with the specific vertex as the [[origin]], then the volume of the parallelepiped is the same as the [[scalar triple product]] of the vectors, or <math>a \cdot (b \times c)</math>.
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The [[volume]] of a parallelepiped is the product of area of one of its faces times the perpendicular distance to the corresponding top face. Alternately, if the three edges of a parallelepiped that meet at one [[vertex]] are defined as [[vector]] <math>a, b,</math> and <math>c</math> with the specific vertex as the [[origin]], then the volume of the parallelepiped is the same as the [[scalar triple product]] of the vectors, or <math>a \cdot (b \times c)</math>.
  
  

Revision as of 20:06, 3 July 2009

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A parallelepiped is a prism that has parallelograms for its faces. Similarly, a parallelepiped is equivalently a hexahedron with six parallelogram faces. Specific parallelepipeds include the cube, the cuboid, and any rectangular prism.

Specific Cases

A parallelepiped with all rectangular faces is a cuboid, and a parallelepiped with six rhombus faces is known as a rhombohedron. In an $n-$dimensional space, a parallelepiped is sometimes referred to as an $n-$dimensional parallelepiped, or as an $n-$parallelepiped. A cube is a parallelepiped with all square faces.

Volume

The volume of a parallelepiped is the product of area of one of its faces times the perpendicular distance to the corresponding top face. Alternately, if the three edges of a parallelepiped that meet at one vertex are defined as vector $a, b,$ and $c$ with the specific vertex as the origin, then the volume of the parallelepiped is the same as the scalar triple product of the vectors, or $a \cdot (b \times c)$.


See also