In three dimensions, a parallelepiped is a prism whose faces are all parallelograms. Let 
, 
,
and 
be the basis vectors defining a three-dimensional parallelepiped. Then the parallelepiped
has volume given by the scalar
triple product
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
In 
dimensions, a parallelepiped is the polytope spanned
by 
vectors 
, ..., 
in a vector space over the
reals,
 |
(4)
|
where ![t_i in [0,1]](/images/equations/Parallelepiped/Inline17.svg)
for 
,
..., 
.
In the usual interpretation, the vector space is
taken as Euclidean space, and the content
of this parallelepiped is given by
 |
(5)
|
where the sign of the determinant is taken to be the "orientation" of the "oriented volume" of the parallelepiped.
Given 
vectors 
,
..., 
in 
-dimensional
space, their convex hull (along with the zero
vector)
 |
(6)
|
is called a parallelepiped, generalizing the notion of a parallelogram, or rather its interior, in the plane. If the number of vectors is equal to the dimension, then
 |
(7)
|
is a square matrix, and the volume of the parallelepiped is given by 
,
where the columns of 
are given by the vectors 
. More generally, a parallelepiped has 
dimensional volume given by 
.
When the vectors are tangent vectors, then the parallelepiped represents an infinitesimal 
-dimensional volume element.
Integrating this volume can give formulas for the volumes of 
-dimensional objects in 
-dimensional space. More intrinsically, the parallelepiped
corresponds to a decomposable element of the exterior algebra 
.
