How to Find Surface Area and Volume of a Prism

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A prism is a three-dimensional figure, which means you can put things inside it. In other words, it has both volume and a surface area. The bases of a prism are always identical polygons. For example, if you have a square as one of the faces, you must have an identical square as the other face.

Examples of prisms

The Volume of a Prism

The volume is the amount of space inside a three-dimensional object. Here you’ll learn how to calculate the volume of prisms that have quadrilaterals as their faces.

Formula

The Volume of a Quadrilateral Prism

V = length \cdot width \cdot height

Volume of quadrilateral prism

Example 1

You can calculate the volume of the quadrilateral prism above by multiplying the length, $4$ cm, by the width, $5$ cm, and the height, $3$ cm:

V = 3 cm \cdot 4 cm \cdot 5 cm = 60 {cm}^{3}

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The Surface Area of a Prism

Now you’ll learn how to calculate the surface area of prisms. The surface area is the sum of all the areas from each face of a prism. So, you can simply calculate the area of each face, and add them all together in the end.

Formula

The surface area of a quadrilateral prism is

2 \cdot the base + 2 \cdot {face}_{1} + 2 \cdot {face}_{2}

Example 2

Example of surface area of quadrilateral prism

You can see in this figure that there are three sets of faces, with each set having two faces that are equal to each other. That means that you only have to find the area of one face from each set, and multiply that answer by 2, to get both their areas. You then add the answers together to find the surface area, like this:

\begin{array}{l} Surface area & = 2 \cdot 4 cm \cdot 3 cm \\ + 2 \cdot 4 cm \cdot 5 cm \\ + 2 \cdot 5 cm \cdot 3 cm \\ = 94 {cm}^{2} \end{array}

\begin{array}{l} Surface area & = 2 \cdot 4 cm \cdot 3 cm + 2 \cdot 4 cm \cdot 5 cm + 2 \cdot 5 cm \cdot 3 cm \\ = 94 {cm}^{2} . \end{array}

Example 3

A prism has a triangle with the area $B = 12 {cm}^{2}$ as its base, and height $h = 5 cm$ . Find the volume of the prism.

You find the volume by multiplying the base area $B$ with by height $h$ :

\begin{array}{l} Volume & = B \cdot h = 12 {cm}^{2} \cdot 5 cm \\ = 60 {cm}^{3} \end{array}

Volume = B \cdot h = 12 {cm}^{2} \cdot 5 cm = 60 {cm}^{3}

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