### What is a Prism?** **

In this guide we’re going to show you how to work out the volume of a prism. Before we get to working out the volume though, it’s important to answer a basic question: what actually is a prism?

Well, a prism is a three dimensional shape which has the same two dimensional shape repeated along its length. Take a look at the examples below. All of these shapes are prisms. Notice how they have the same two dimensional shape (a rectangle, triangle or trapezium in these examples) repeated throughout their length. We will call this two dimensional shape the cross-section.

### The Formula for the Volume of a Prism

You may see various formulas for working out the volume of different prisms, but they all follow this same rule:

**Volume of a prism = Area of the Cross-Section x Length**

The cross-section is the two dimensional shape repeated throughout the prism’s length. You simply work out the area of this cross-sectional shape (using your knowledge on area) and then multiply that area by the length (sometimes referred to as the depth). Check out these examples below to help build your understanding.

### Examples

#### Example 1 – Cuboid

Let’s start by working out the volume of this cuboid.

This is a type of prism, with a rectangle repeated throughout the shape’s length. Here’s how we should approach the question and set out our working:

Volume of prism = Area of cross-section x Length

Volume of example 1 = Area and rectangle x Length

= Base x Height x Length

= 8 x 3 x 10

= 240cm^{3}

#### Example 2 – Triangular Prism

Work out the volume of this triangular prism.

This type of prism has a triangle as the cross-section, so we’re going to work out the area of the triangle (check out our area guide for help here) and then multiply that by the length.

Volume of prism = Area of cross-section x length

Volume of example 2 = Area of triangle x depth

= (1/2 x base x height) x length

= (1/2 x 6 x 4) x 9

= 12 x 9

= 108cm^{3}

#### Example 3 – Trapezoidal Prism

Work out the volume of this trapezoidal prism.

Here we have a prism with a trapezium as the cross-section. We’re going to use the same formula for working out the volume of a prism. We’ll find the area of the cross-section (which is a trapezium in this example) and then multiply by the length to work out the volume. Again, if you need to revise the area of a trapezium then you should check our revision guide on area as well.

Volume of prism = Area of cross-section x length

Volume of example 3 = Area of trapezium x length

= 0.5 x (a + b) x h x length

= 0.5 x (7 + 9) x 5 x 8

= 40 x 8

= 320 cm^{3}

That is pretty much everything you need to know on this topic. Firstly, work out the area of the cross-section. Then simply multiply by the length and you have your answer. Check out some more great support on this topic from BBC bitesize to help you practice.

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