Volume of Parallelepiped

In mathematical geometry, a parallelepiped is defined as the 3-D figure that is formed by the six parallelograms together. Sometimes, the term rhomboid is also defined with the same meaning. A parallelepiped is related to the parallelogram in the same manner how a cube related to the square and a cuboid related to the rectangle.

Here are given three equivalent definitions of the parallelepiped to understand the concept deeply –

  • This could be defined as the polyhedron with six faces (hexahedron) where each of the sides is defined as the parallelogram.
  • It could be given as a hexahedron where three pairs for parallel faces and
  • It could be defined as the prism too where the base is a parallelogram.

The specific cases of a parallelepiped could be given as –

  • The rectangular cuboid (six rectangular faces),
  • cube (six square faces), and
  • the rhombohedron (six rhombus faces)

In general, a parallelepiped is given as 3 sets of 4 parallel edges where edges within each set are of equal length. This geometric figure is the result of a linear transformation for a cube. As each face has to give point symmetry then a parallelepiped could be defined as the zonohedron too where each face is seen from the outside as a mirror image of the opposite face.

You must be wondering how to calculate the volume of a parallelepiped when the area of base and height is given. In that case, you just need to multiply height and base together. The base is the length of faces and height is the perpendicular distance between the base and opposite side.

The volume of Parallelepiped Formula

Mathematically, The volume of the parallelepiped equals the absolute value of the scalar triple product, a · (b × c):

\[\large volume\;of\;Parallelepiped =S\times h\]

S= Area of the bottom
h= Height

\[\large volume\;of\;Parallelepiped = a(b\times c)=b(c\times a) = c(b\times a)\]