Math resources Geometry Surface area

Surface area of a cube

# Surface area of a cube

Here you will learn about the surface area of a cube, including how to calculate the surface area of a cube and how to find missing values of a cube given its surface area.

Students will first learn about the surface area of a cube as a part of geometry in 8 th grade.

## What is the surface area of a cube?

The surface area of a cube (cuboid) is the sum of the areas of all the faces of a cube. This is also referred to as the total surface area of a cube (tsa) or the lateral surface area of a cube.

A cube is a three-dimensional figure that has six congruent square faces. This means that all the side faces are the same size.

To find the area of each face, multiply the side lengths together. Then multiply the area of each of the square faces by six.

The formula to calculate the surface area, S, of a cube is S=6x^{2} where x represents the side length (edge length) of the cube.

The surface area of a cube formula can be used to find the surface area of any cube.

Surface area is measured in square units, for example mm^{2}, \, cm^{2} or \, m^{2}.

## Common Core State Standards

How does this relate to 8 th grade math?

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

## How to calculate the surface area of a cube

In order to calculate the surface area of a cube:

1. Write the formula for the surface area of the cube.
2. Substitute any known value(s) into the formula.
3. Complete the calculation.
4. Write the solution, including the units.

## Surface area of a cube examples

### Example 1: surface area of a cube (integer side lengths)

Find the surface area of the cube below.

1. Write the formula for the surface area of the cube.

S=6x^{2}

2Substitute any known value(s) into the formula.

Here, x=5 and so S=6\times{5}^{2}.

3Complete the calculation.

S=6\times{5}^{2}=6\times{25}=150

4Write the solution, including the units.

As the unit of length is centimeters (cm), the unit of area is square centimeters (cm^{2}).

S=150 \, cm^{2}

### Example 2: surface area of a cube (one known edge of the cube)

Find the surface area of the cube.

Write the formula for the surface area of the cube.

Substitute any known value(s) into the formula.

Complete the calculation.

Write the solution, including the units.

### Example 3: surface area of a cube (worded problem)

A cube structure has a side length of 7 \, m. Calculate the total surface area of the structure.

Write the formula for the surface area of the cube.

Substitute any known value(s) into the formula.

Complete the calculation.

Write the solution, including the units.

### Example 4: surface area of a cube (area of a face given)

The area of the face of a cube is 30 \, cm^{2}. Find the surface area of the cube.

Write the formula for the surface area of the cube.

Substitute any known value(s) into the formula.

Complete the calculation.

Write the solution, including the units.

### Example 5: find the length of a cube given the surface area

The surface area of a cube is 24 \, ft^{2}. Find the length of the cube.

Write the formula for the surface area of the cube.

Substitute any known value(s) into the formula.

Complete the calculation.

Write the solution, including the units.

### Example 6: find the length of a cube given the surface area (decimal solution)

The surface area of a cube is 483 \, mm^{2}. Find the length of the side x correct to 2 decimal places.

Write the formula for the surface area of the cube.

Substitute any known value(s) into the formula.

Complete the calculation.

Write the solution, including the units.

### Teaching tips for surface area of a cube

• When introducing the surface area of a cube, use a math net to break down the cube into individual cubes, showing that each side of a cube is a square.

• Allow students time to work through a variety of problems, on worksheets or using a technology tool. Students should have time to work independently, as well as with other students, to really reinforce learning and understanding.

• Use visuals and real life examples of cubes, such as a rubik’s cube, to allow students to physically hold and explore cubes.

### Easy mistakes to make

• Missing/incorrect units
Surface area is measured in square units (for example, \mathrm{mm}^2, \mathrm{~cm}^2, \mathrm{~m}^2 etc.).

• Calculating with different units
You need to make sure all measurements are in the same units before calculating the surface area. For example, you can’t have some in cm and some in m.

• Calculating volume instead of surface area
Volume and surface area are different quantities. The volume of the cube is the three-dimensional space in a shape and is measured in cubic units. Surface area is the amount of space covering the outside of a 3D shape. To find surface area, find the area of one face and multiply it by six.

### Practice surface area of a cube questions

1. Find the surface area of the cube.

27 \, cm^3

12 \, cm^2

36 \, cm^2

54 \, cm^2

S=6x^{2} where x=3 \, cm.

S=6\times{3}^{2}=6\times{9}=54 \, cm^{2}.

2. Calculate the surface area of the cube below. Write your answer in square centimeters.

0.125 \, cm^3

15,000 \, cm^2

6 \, cm^2

1.5 \, cm^2

S=6x^{2} where x=0.5 \mathrm{~m}=50 \mathrm{~cm}.

S=6\times{50}^{2}=6\times{2500}=15,000 \, cm^{2}.

3. Find the surface area of the cube. Give your answer in cm^2.

96 \, cm^2

9,600 \, cm^2

960 \, cm^2

9.6 \, cm^2

S=6x^{2} where x=40 \mathrm{~cm}=0.4 \mathrm{~m}.

S=6\times{40}^{2}=6\times{1600}=9,600 \, cm^{2}.

4. The surface area of a cube is 150 \, cm^2. Find the length of the side of the cube.

135,000 \, cm

2.04 \, cm

5 \, cm

12.5 \, cm

S=6x^{2} where S=150 \, cm^{2}.

\begin{aligned}150&=6\times{x}^{2} \\\\ 25&=x^{2} \\\\ x&=\sqrt{25} \\\\ x&=5cm \end{aligned}

5. The surface area of the cube is 6 \, m^2. Find the length of each side x.

2 \, m

1 \, m

0.3 \, m

0.41 \, m

S=6x^{2} where S=6 \, m^{2}.

\begin{aligned}6&=6\times{x}^{2} \\\\ 6\div{6}&=x^{2} \\\\ x^{2}&=1 \\\\ x&=\sqrt{1} \\\\ x&=1m \end{aligned}

6. The surface area of a cube is 186 \, m^2. Find the length of each side. Write your answer to the nearest centimeter.

5.57 \, m

2.27 \, m

15.50 \, m

207,576 \, m

S=6x^{2} where S=186 \, m^{2}.

\begin{aligned}186&=6\times{x}^{2} \\\\ 31&=x^{2} \\\\ x&=\sqrt{31}=5.567764362830…=5.57 \, (2 d p). \end{aligned}

## Surface area of a cube FAQs

What is a cube?

A cube is a three-dimensional figure with six equal square faces and 8 vertices.

What is the formula to find the volume of a cube?

To find the volume of a cube, you will use the formula V=a^3 or
V=\text { side } \times \text { side } \times \text { side }.

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