What is a square number? Here’s everything you and your child needs to know about square numbers in KS2 primary school maths. We will look at what a square number is, how you should square a number, why they are called square numbers and how to calculate the inverse.

This is how to help your child fully get to grips with square numbers. Moreover, we’ve included a square numbers worksheet with fun practice examples and solutions.

What is a square number?

A square number is a number multiplied by itself. It can also be called ‘a number squared,’ and all square numbers are formed by finding the product of an integer with itself.

In mathematics, the symbol used to represent any number squared is written in superscript, like this:  ²  for example 3² = 3 x 3 = 9.

Mathematically speaking, a square number is a number of the form n x n, or n² where n is any integer (an integer is any whole number including 0 and negative numbers).

To understand this better, we will take a look at an unknown variable c (we don’t know what c is, but that doesn’t matter…)

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

c³ is ‘c to the power of 3’  most commonly known as ‘c cubed’

c⁴ is ‘c to the power of 4’

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 squares, the result of a number raised to the power of 2.

Why is a square number called a ‘square number’?

The real life representation of multiplying a number by itself or squaring a number, for example 4², is calculating the area of a square with side length 4. These numbers are called square numbers for the simple reason that they form the area of a square.

Let’s take a look at the most basic square, the unit square. The area of a rectangle is given by the formula:

 Area = length x width

It is easy to see that using the formula, the area of the unit square is 1 x 1 = 1 square unit. In fact because squares have equal sides, you just need to pick one side and ‘square’ it.

In this example a square of width 4cm is made up of 4 rows and 4 columns of 1cm² squares. To calculate the area, you can count the total number of squares inside the square, each having an area of 1 square centimeter.

In this case the area of the square is 4cm x 4cm = 16cm² or 16 square centimeters.

What are the square numbers up to 1000

0² = 0 x 0 = 0                 10² = 10 x 10 = 100                   21² = 21 x 21 = 441

1² = 1 x 1 = 1                 11² = 11 x 11 = 121                   22² = 22 x 22 = 484

2² = 2 x 2 = 4                 12² = 12 x 12 = 144                   23² = 23 x 23 = 529

3² = 3 x 3 = 9                  13² = 13 x 13 = 169                   24² = 24 x 24 = 576

4² = 4 x 4 = 16               14² = 14 x 14 = 196                   25² = 25 x 25 = 625

5² = 5 x 5 = 25               15² = 15 x 15 = 225                   26² = 26 x 26 = 676

6² = 6 x 6 = 36               16² = 16 x 16 = 256                   27² = 27 x 27 = 729

7² = 7 x 7 = 49               17² = 17 x 17 = 289                   28² = 28 x 28 = 784

8² = 8 x 8 = 64               18² = 18 x 18 = 324                   29² = 29 x 29 = 841

9² = 9 x 9 = 81               19² = 19 x 19 = 361                   30² = 30 x 30 = 900

20² = 20 x 20 = 400                   31² = 31 x 31 = 961

Learning square numbers

This list does not contain all of the square numbers; any number can be squared so the list could go on forever. You are also not expected to know all of these numbers in KS2 primary school maths, so do not worry!

What is the square root?

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

For example: 3 squared = 9, and so the square root of 9 is equal to 3

Primary school children may 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 area of of a square then in order to work out the length of the sides, you must calculate the square root. Here are some examples of square roots for the first square numbers up to 20:

1² = 1 x 1 = 1                                        square root of 1 = 1

2² = 2 x 2 = 4                                        square root of 4 = 2

3² = 3 x 3 = 9                                        square root of 9 = 3

4² = 4 x 4 = 16                                     square root of 16 = 4

Question: The area of a square is 25cm², calculate the side length?

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

We need to consider what number when multiplied by itself gives us 25.

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

In this case, it is easy to spot that 25 = 5 x 5 = 5² which gives us the dimensions of the square.

Length = 5cm

Width = 5cm

When will my child learn about square numbers in school?

It is not until Year 5 maths classes that students will encounter squaring numbers. From Year 4, students are taught to recall multiplication and division facts for multiplication tables up to 12 x 12 so your child may already be familiar with some the square numbers up to 144, and should definitely recognize the square numbers up to 50.

The national curriculum states that a Year 5 maths student should be taught to:

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

The non-statutory notes and guidance state that Year 5 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.

This learning will be consolidated in Year 6, particularly when learning about the order of operations where they will be introduced to the mathematical term ‘index.’ They will also begin to perform mental calculations with higher numbers, increasing their fluency through the use of multiplication tables.

Square number practice questions

• What are the first 5 square numbers?
• If the length of a square is 6cm, what is the area of the square?
• Explain why 1 is a square number?
• A square has an area of 64mm², what is the length of the square?
• Fill in the table:

• Bonus questions:

This question was taken from the 2019 KS2 maths SATs exam, testing children’s application of their knowledge of square numbers

• Another question on area taken from the 2019 KS2 Mathematics reasoning paper

Solutions

• 0, 1, 4, 9, 16
• 6cm x 6cm = 36 cm²
• 1 is a square number because it is the product/ result of multiplying 1 with itself 1 x 1 = 1. 1 square unit is the area of the unit square
• 64 = 8 x 8 so length = 8mm
• There are many solutions to this problem. The first column must contain two even numbers (ending in 0, 2, 4, 6 or 8), the first being a square number and the second not.

The second column must contain two odd numbers (ending in 1, 3, 5, 7, or 9)

E.g

6) Rectangle divided as shown (different orientations also work)

7)    Perimeter of hexagon = 8 x 6 = 48 (There are six sides each with length 8cm).

A square with perimeter 48 has sides of length 48/4 = 12cm

So area of square = 12²  = 144cm²

You may also like to read:

Your Must Have Guide to KS2 Fractions

What are Cube Numbers? How to Explain Cube Numbers to Primary School Students?

Why Online Maths Tutors are More Popular than Ever in 2020?