Surface Area of a Box

A box, or cuboid, is a 3-dimensional shape made up of 6 faces, all of which are rectangles.

Each face of the box has an opposite identical face which we can see clearly if we look at the net.

So a box has 3 separate pairs of identical faces.

The surface area of the box is equal to the total of all the areas of the rectangular faces.

So the total surface area of the box is: \[ A = lw + lw + lh + lh + wh + wh = 2lw + 2lh + 2wh. \]

Surface Area of a Box Example 1

Find the surface area of the box below.

The box has the following dimensions:

• Length: 7 cm
• Width: 3 cm
• Height: 5 cm

The surface area of the box is:

\[ A = 2lw + 2lh + 2wh \; \]

where l is the length, w is the width, and h is the height of the box

So if we substitute the values of the length and width into this equation, we get: \[ A = 2 \cdot 7 \cdot 3 + 2 \cdot 7 \cdot 5 + 2 \cdot 3 \cdot 5 = 42 + 70 + 30 = 142 \]

The surface area of the box is 142 cm².

Surface Area of a Box Example 2

Find the surface area of the box below.

The box has the following dimensions:

• Length (l):12 in
• Width (w): 5 in
• Height (h): 3 in

The surface area of the box is:

\[ A = 2lw + 2lh + 2wh \; \]

So if we substitute the values of the length and width into this equation, we get: \[ A = 2 \cdot 12 \cdot 5 + 2 \cdot 12 \cdot 3 + 2 \cdot 5 \cdot 3 = 120 + 72 + 30 = 222 \]

The surface area of the box is 222 in².

Surface Area of a Box Example 3

Find the surface area of the box below, giving your answer to 1 decimal place.

The box has the following dimensions:

• Length (l): 5.6 cm
• Width (w): 4.3 cm
• Height (h): 2.5 cm

The surface area of the box is:

\[ A = 2lw + 2lh + 2wh \; \]

So if we substitute the values of the length and width into this equation, we get: \[ A = 2 \cdot 5.6 \cdot 4.3 + 2 \cdot 5.6 \cdot 2.5 + 2 \cdot 4.3 \cdot 2.5 = 48.16 + 28 + 21.5 = 97.66 \]

The surface area of the box is 97.7 cm² to 1 decimal place.

Surface Area of a Box Example 4

Find the surface area of the open box below.

The box has the following dimensions:

• Length (l): 6 in
• Width (w): 3 ½ in
• Height (h): 2 ½ in

The formula for the surface area of an open box is:

\[ A = lw + 2lh + 2wh \; \]

So if we substitute the values of the length and width into this equation, we get: \[ A = 6 \cdot 3 {1 \over 2} + 2 \cdot 6 \cdot 2 {1 \over 2} + 2 \cdot 3 {1 \over 2} \cdot 2 {1 \over 2} = 21 + 30 + 17 {1 \over 2} = 68 {1 \over 2} \]

The surface area of the box is 68 ½ in².