# Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms

Three equivalent definitions of parallelepiped are:

• a polyhedron with six faces (hexahedron), each of which is a parallelogram,
• a hexahedron with three pairs of parallel faces, and
• a prism of which the base is a parallelogram.

Right parallelogrammic prism has four rectangular faces and two parallelogrammic faces and it is a special case of parallelepiped.

## Side area of a right parallelogrammic prism

$$S_{k}=2H(a+b)=C\times H,$$

where,

a,b— lenghts of sides;
C— circumference of bottom side;
H— height.

## Base area of a right parallelogrammic prism

\begin{align} S_{p}&=ab \sin \alpha = ah_{a},\\ \end{align}

where,

a,b— lenghts of sides;
α— the acute angle between the base and the height of the bottom;
ha— height of the base.

## Area of a right parallelogrammic prism

\begin{align} S_{t}&=S_{k}+2S_{p},\\ \\ S_{t}&=2H(a+b) + 2ab \sin \alpha = 2H(a+b) + 2ah_{a},\\ \end{align}

where,

a,b— lenghts of sides;
α— the acute angle between the base and the height of the bottom;
ha— height of the base.

## Volume of a right parallelogrammic prism

$$V=S_{p}\times H,$$