Beside

Square Numbers

In one line: A square number is the answer you get when you multiply a whole number by itself, like 4 × 4 = 16.

Why it works

Square numbers come up all over maths. You use them to find the area of squares, to work with powers (a short way of writing repeated multiplication), and later to find square roots and use Pythagoras' theorem. Knowing the first twelve or so by heart makes mental maths faster. It also helps you spot patterns and check whether an answer is sensible.

Think of it like: Imagine you have a pile of square tiles and you want to lay them out as a perfect square, with the same number of tiles in every row and every column. With 9 tiles you can make a 3-by-3 square, so 9 is a square number. With 10 tiles you always have some left over or a gap, so 10 is not. That is where the name comes from: a square number of tiles makes a square. We write 3 × 3 as 3², read as "3 squared". The small 2 tells you to multiply the number by itself, not by 2. (Where this breaks: The tile picture only works for whole numbers. You can also square fractions, decimals and negative numbers (for example, 1.5² = 2.25 and (−3)² = 9), but you cannot lay out 2.25 tiles as a neat square. When people say "square numbers", they usually mean the squares of whole numbers: 1, 4, 9, 16, 25 and so on.)

Worked example

Which numbers between 20 and 70 are square numbers?

  1. Start squaring whole numbers in order: 1² = 1, 2² = 4, 3² = 9, 4² = 16.
  2. 4² = 16 is still below 20, so keep going.
  3. 5² = 25, 6² = 36, 7² = 49 and 8² = 64 all lie between 20 and 70.
  4. 9² = 81 is bigger than 70, so you can stop here.

Answer: The square numbers between 20 and 70 are 25, 36, 49 and 64.

Common mistakes

  • Reading 5² as 5 × 2 = 10 — the small 2 means multiply 5 by itself, so 5² = 5 × 5 = 25.
  • Thinking square numbers must be even — the list goes odd, even, odd, even (1, 4, 9, 16, 25…), so check by finding a whole number that multiplies by itself to make it.
  • Mixing up squares and cubes — squaring means multiplying the number by itself once (4² = 4 × 4 = 16), while cubing uses it three times (4³ = 4 × 4 × 4 = 64).
  • Confusing the area of a square with its perimeter (the distance all the way round) — a square with sides of 6 cm has an area of 6² = 36 cm² but a perimeter of 4 × 6 = 24 cm.

Try it · 1 of 3

Free · no sign-up
What is 7²?

Questions parents ask

What year do children learn square numbers?

In the UK, square numbers and the ² notation are introduced in Year 5 and built on in Years 6 and 7. In Australia they usually appear around Years 5–7, with square roots in Year 7. In the US, children meet them around Grades 5–6, when exponents are taught, and study square roots more formally in Grade 8.

How can I help my child learn square numbers at home?

Use counters, coins or squared paper to build squares (2 by 2, 3 by 3 and so on) and count the total each time. Then practise reciting the squares up to 12 × 12, and look for patterns, such as how the gaps between them go 3, 5, 7, 9.

What is the difference between a square number and a square root?

A square number is the answer to multiplying a number by itself, such as 6 × 6 = 36. The square root works backwards: it asks which number multiplied by itself gives 36, so the square root of 36 is 6.

Are 0 and 1 square numbers?

Yes, 1 is a square number because 1 × 1 = 1, and it is usually the first one children learn. 0 is also the square of a whole number (0 × 0 = 0), so many mathematicians count it, although some school lists start at 1.