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;

h_{a}— 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;

h_{a}— height of the base.

## Volume of a right parallelogrammic prism