Geometry

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# Volume of Parallelepiped

#### Lesson Contents

## Parallelepiped Volume Formula

There are two formulas for finding volume of a parallelepiped. They are given as:**V = h·|a|·|b|·sin(γ)****V = h·B**

Where *V* is the volume, *h* is the height, *a* and *b* are the base edge vectors, *γ* is the angle between vectors *a* and *b*, and *B* is the area of the base.

### What is a Parallelepiped?

A parallelepiped is a three-dimensional shape made of 6 faces. It is the result of tilting the edges of a rectangular prism. Imagine pushing against the top corner of a box that is not perfectly rigid. The box will slant in the direction that it is pushed. This forms a parallelepiped.

As we can see in the image above, there are three pairs of congruent parallelograms on opposing sides of the figure. This is the most common style of parallelepiped. **However, not all parallelepiped shapes have three pairs of opposing congruent sides.**

It is easier to calculate the volume of parallelepiped type shapes if we understand that a parallelepiped is formed by six parallelograms. If we understand how to calculate volume of a rectangular prism and can visualize what a parallelepiped is, we need not memorize the formula. Finding the area of the base parallelogram and multiplying by the shape’s height will give us the volume.

### Example Problem

A given parallelepiped is made up of 3 pairs of congruent parallelograms. The base parallelogram has an area of 8. The height of the parallelepiped is 4. What is its volume?

Solution:

1.) Since we are given base area and height, we can use the simplified formula V = h∙B.

2.) Let’s plug the base area and height into the formula.

V = (4)(8) = 32.

3.) **The volume of the parallelepiped is 32.**