How to figure out the total surface area of a cuboid (rectangular prism).

Working out the surface area of a cuboid (rectangular prism) can be done by working out the total area of all of the six faces. Since all the faces are rectangular you will need to know how to find the area of a rectangle – this can be done by multiplying the two side lengths of the rectangle together (length × width). Let’s look at a couple of examples on calculating the surface area of a cuboid.

Example 1

Work out the surface area a cuboid that is 7cm by 6cm by 5cm.

All you have to do then is work out the total area of all of the faces. The faces have been coloured in to make the surface area easier to calculate.

First let’s work out the area of the green face. The two sides of the green face are 7cm and 5cm so the area of the green face is:

7 × 5 = 35 cm²

Next, work out the area of red face. The two sides of the red face are 6cm and 5cm so the area of the red face is:

6 × 5 = 30 cm²

Finally, work out the area of the blue face. The two sides of the blue face are 7cm and 6cm so the area of the blue face is:

7 × 6 = 42 cm²

Now there are 3 more faces which have the same area of the 3 faces above - so if you add all these up you will get the total surface area of the cuboid:

35 + 35 + 30 + 30 + 42 + 42 = 214 cm²

Let’s look at one more example.

Example 2

A rectangular box measures 38cm by 26cm by 70cm. All of the 6 faces needed to be painted. One tube of paint will cover 500cm². Work out how many tubes of paint need to be bought to cover the box in paint.

To answer this question you will need to figure out the surface area of the cuboid.

First let’s work out the area of the green face. The two sides of the green face are 7cm and 5cm so the area of the green face is:

70 × 38 = 2660 cm²

Next, work out the area of red face. The two sides of the red face are 26cm and 38cm so the area of the red face is:

26 × 38 = 988 cm²

Finally, work out the area of the blue face. The two sides of the blue face are 70cm and 26cm so the area of the blue face is:

70 × 26 = 420 cm²

Now there are 3 more faces which have the same area of the 3 faces above, so if we add all these up you will get the total surface area of the cuboid:

2660 + 2660 + 988 + 988 + 420 + 420 = 8136 cm²

All you need to do now is work out the amount of tubes of paint you will need to cover the box. This can be done by dividing the total surface area by 500 (since 1 tube covers 500 cm²)

So 8136 ÷ 500 = 16.272

So you will need 17 tubes of paint.