Surface Area of a Box

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.

Surface Area of a Box Calculator

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

Surface Area of a Box Calculator

surface area of a box labelled image


Answer


Answer

 
 


How the Calculator Works

  1. 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).
  2. Choose your units of measurement (default is none)
  3. Choose your desired accuracy (default is 2 decimal places)
  4. Click the Find Surface Area button
  5. 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.

Surface Area of a Cuboid Worksheets

We have created two worksheets for you to practice this skill.

The first sheet involves finding the surface area of different cuboids with simple numerical values for the side measurements.

The second sheet is the same as the first but has decimal and fractional values for the side measurements.

More Recommended Math Resources

Take a look at some more of our worksheets similar to these.

More Surface Area Worksheets and Support Pages

More Area Support and Resources

We have a range of other area worksheets and support pages for a range of different 2d shapes.

More Area & Volume Calculators

We have a range of area and volume calculators to help you find the area and volumes of a range of different 2d and 3d shapes.

Each calculator page comes with worked examples, formulas and practice worksheets.

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Our calculator hub page contains links to all of our calculators!

 

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