We’ve all heard of cube numbers, but what exactly are cube numbers? If you find yourself pulling out your hair during homework time, we’ve broken it down for you. Here’s how to help your child better understand cube numbers at Key Stage 2 level primary school maths.

We will look at what cube numbers are, why they are called cube numbers, how to calculate the inverse, and we will give you all the information required to help your child get up to speed on this topic. There are also some fun worked examples with solutions to try at the end!

What is a cube number in maths?

Cube numbers are the product of three identical numbers. Basically, you need to multiply a number by itself, and then by itself again! This can also be called ‘a number cubed.’

In maths, the symbol used to represent any number cubed is written in something called – superscript. Here’s an example:  ³ for example 3³ = 3 x 3 x 3 = 27.

To understand this better, we will use unknown variable c (we don’t know what c is yet, but it doesn’t matter…)

First consider, c²  is ‘c  to the power of 2’  most commonly known as ‘c squared’

It’s good to note that c³ is ‘c to the power of 3’ most commonly known as ‘c cubed’

Then, c⁴ is ‘c to the power of 4’

Finally, c⁵ is ‘c to the power of 5’

A number can be raised to any power (or index) that you like, however today we will focus on just the cubes, the result of a number raised to the power of 3.

You may have already noticed that a number cubed is the same as multiplying the original number by its square number.

2 squared = 2 x 2 = 4

2 cubed = 2 x 2 squared = (2 x 2) x 2 = 8

For each time you multiply a number by itself you must raise the power by 1.

Why are cube numbers called cube numbers?

The real-life representation of multiplying a number by itself twice over, for example 4 x 4 x 4, is finding the volume of a cube with side length 4.

So in this example a square of length 4cm is made up of 4 rows and 4 columns of 1cm² squares, and to calculate the area you must count the total amount of squares, each with an area of 1cm²

In this case area of the square = 4 x 4 = 4² = 16 so we call this 4 ‘squared’

We are now going to use the exact same method to calculate the volume of a cube.

This time, the shape has another dimension, and now we must also consider the depth. Volume means the amount of 3-dimensional space a shape takes up, so not only do we need the area of one face, we need the total amount of 1cm³ blocks.

Now there are 4 x 4 x 4 little cubes, giving us a volume of 64cm³

A cube by definition has equal height, width, and depth, and so to calculate the volume of a cube you must multiply these dimensions together. This is where the name cube comes from. Cube numbers can be used to calculate the volume of a cube.

What are the cube numbers from 1-100?

0³ = 0 x 0 x 0 = 0

1³ = 1 x 1 x 1 = 1

2³ = 2 x 2 x 2 = 8

3³ = 3 x 3 x 3 = 27

4³ = 4 x 4 x 4 = 64

5³ = 5 x 5 x 5 = 125

6³ = 6 x 6 x 6 = 216

7³ = 7 x 7 x 7 = 343

What is a cube root?

To find the cube root of a number you must find some number that when multiplied by itself twice gives you the original number you started with. The cube root is the inverse function of cubing a number.

Cube root examples:

For example: 2 cubed = 8, and so the cube root of 8 is equal to 2

You will have already encountered the inverse functions of addition, subtraction, multiplication and division as the opposite operation, used to reverse a calculation that you have just done (or to check your work). This inverse function is very important because if you know the volume of a cube, then in order to work out the dimensions of the cube, you must calculate the cube root.

1³ = 1 x 1 x 1 = 1                                cube root of 1 = 1

2³ = 2 x 2 x 2 = 8                                 cube root of 8 = 2

3³ = 3 x 3 x 3 = 27           ——–>          cube root of 27 = 3

4³ = 4 x 4 x 4 = 64                               cube root of 64 = 4

5³ = 5 x 5 x 5 = 125                            cube root of 125 = 5

Question: The volume of a cube is 27m³, calculate the height?

In this example here, we are given the volume and asked to work backwards. Alarm bells should be ringing here that we need to use the cube root.

We need to consider what number when multiplied by itself 2 consecutive times gives us 27.

In KS2 primary school maths, you are not expected to know all of the cube numbers off by heart, but it will definitely help to recognise the first few.

In this case, it is easy to spot that 27 = 3 x 3 x 3 = 3³ which gives us the dimensions of the cube.

Length = 3m

Height = 3m

Depth = 3m

What age will my child start learning about cube numbers?

Children are first introduced to square numbers to calculate the area of a square in Year 5. They will be expected to recognise the notation ² and to be able to calculate the area of basic squares and rectangles. They will also begin to estimate the area of shapes by counting how many 1cm² squares make up the shape.

The Year 5 national curriculum for maths indicates that pupils should:

• recognize and use square numbers and cube numbers, and the notation for squared (²) and cubed (³)
• solve problems involving multiplication and division, including using their knowledge of factors and multiples, squares and cubes

The non-statutory notes and guidance state that Y5 pupils should be able to:

• Use and understand the terms factor, multiple and prime, square and cube numbers
• Use them to construct equivalence statements (for example, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 9² x 10

In Year 5 they will have a brief introduction to cube numbers and begin to learn about the capacity of 3 dimensional objects, applying the same theory they learnt about for square numbers, this time using 1cm³ building blocks to estimate the volume of a cuboid. They should be familiar with some of the different units, i.e cm³, m³ and mm³ (cubic centimetres, cubic metres and cubic millimetres respectively). Consolidation of this will be in Year 6.

Often in 11+ exams and secondary school entrance exams students will be given a pattern of squared or cubed numbers and be expected to spot the pattern and continue it.

What is a perfect cube?

When we talk about cube numbers, we usually mean perfect cube numbers. A perfect cube is an integer (or whole number) that can be written as another integer to the power of 3.

For example, 125 is a perfect cube, because it is the product of an integer multiplied by itself 2 times, 5 x 5 x 5 = 125

Using a calculator we can see 121 is not a perfect cube because there doesn’t exist an integer that when cubed gives the value 121. Using a calculator we can see ∛121=4.94608744. This means that the cube root of 121 is approximately 4.95

This is a scary looking number and in KS2 primary school maths the only cube numbers you only need to be able to recognize are the nicely fitting, perfect cube numbers.

Cube numbers: Examples and practice questions

1. What is the cube root of 64?
2. A dice has volume 216mm³, what is the area of one face?
3. Order these from smallest to largest: 5², 3², 3³, 2³
4. Find two cube numbers that total 3²
5. Use long multiplication to calculate 7³

Solutions

1. ∛64 = ∛(4 x 4 x 4) = 4
2. We must spot that 216 = 6³ so the dice has measurements 6mm x 6mm x 6mm. The length of every edge in a cube is the same so to calculate the area of a face we just need to multiply 6mm x 6mm = 36mm² (take note of how the units become mm² now we are calculating area instead of volume.
3. 2³ = 8 | 3² = 9 |5² = 25 |3³ = 27
4. 3² = 9 The only solution to this equation using perfect cubes is 2³ + 1³
5. 7³ = 7 x 7² Can spot first that 7² is 49? Then long multiplication used to calculate 49 x 7.

You may also like to read:

How to Work Out Area of Different Shapes at KS2

The KS2 Long Multiplication Method – The Step by Step Guide for Parents and Kids

What is BODMAS? A Parent’s Guide to BODMAS, BIDMAS, and The Order of Operations