Surface Area of a Box

$Surface Area of a Box Calculator Picture$

Welcome to our Surface Area of a Box page.

We explain how to find the surface area of a cuboid, or box, and provide a quick calculator to work it out for you, step-by-step.

Our calculator will find the surface area of both closed and open boxes.

There are also lots of worked examples and some worksheets for you to practice this skill.

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Surface Area of a Box Calculator
Surface Area of a Cuboid / Box Formula
Surface Area of a Cuboid / Box Explained
Surface Area of a Box Examples
Surface Area of a Cuboid Worksheets
More recommended resources

Surface Area of a Box Calculator

This calculator finds the surface area of a cuboid (or box) when the side lengths are known.

How the Calculator Works

Choose the length, width and height of the rectangle: you can choose a whole number, decimal or fraction.
- You can type a fraction by typing the numerator then '/' then the denominator.
- You can type a mixed number by typing the whole-number part, then a space then the fraction part.
- Examples: 2 1/2 (two and one-half); 3 4/5 (three and four-fifths); 7 1/3 (seven and one-third).
Choose your units of measurement (default is none)
Choose your desired accuracy (default is 2 decimal places)
Click the Find Surface Area button
You will be shown the area as a decimal (and also a fraction if you typed the length as a fraction).

Surface Area of a Box (or Cuboid) Formula

Below are the formulas for the surface area of both a closed box and an open box.

An open box is basically a normal box which does not have a lid so it is open at the top.

$surface area of a box formula image$

The formula for the surface area of a closed box (or cuboid) is: \[ A = 2lw + 2lh + 2wh \; \]

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

The formula for the surface area of an open box (with no lid) is: \[ A = lw + 2lh + 2wh \; \]

Surface Area of a Box Explained

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

$area of a box explained image$

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.

$area of a box explained image 2$

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 Examples

Surface Area of a Box Example 1

Find the surface area of the box below.

$area of a box example 1$

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.

$area of a box example 2$

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.

$area of a box example 3$

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.

$area of a box example 4$

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.