Geometry Help

Volume of Parallelepiped

A parallelepiped is a very tongue twisted name for a three dimensional square, rectangle or parallelogram. Any cardboard box you see around is most likely a parallelepiped. If the box is upright, in another word, the sides are perpendicular to the base of the box, then the mathematical term for that box is rectangular parallelepiped. So, before we calculate the volume of a parallelepiped, let us examine closely the differences between a rectangular parallelepiped and other parallelepiped, as well as what a parallelepiped actually is.


This parcel cardboard box is an example of a parallelepiped. Because the box is upright and the sides are perpendicular to the base, it is also an example of a rectangular parallelepiped. Let's examine its geographical shape in more detail.

Rectangular parallelepiped

Rectangular parallelepiped

The rectangular parallelepiped is a 3 - D version of a rectangle or a square where b is perpendicular to a. A rectangular parallelepiped is made up of 6 rectangles. Now we compare this rectangular parallelepiped to other types of parallelepipeds.

Volume of parallelepiped

Only rectangular parallelepiped has perpendicular sides. Other parallelepiped can have slanted sides. In another word, a parallelepiped is made up of six parallelograms where a, b, and c are unique dimensions. A in the diagram represents the area of a side of the parallelepiped which we are going to use to calculate the volume. If you do not know the area A, then you can use the formula for the area of a parallelogram to calculate A first. Just to recap, the area of a parallelogram is just its base times perpendicular height.

Another measurement we need to make before we can calculate the volume of the parallelepiped is the height between A and its parallel side. The height is given by h in the diagram. Note that h has to be perpendicular to the side which you just calculated the area of. It cannot be any slanted side of the parallelepiped.

Now we are ready to calculate the volume of a parallelepiped using this formula.

Parallelepiped formula