Square and Cube Numbers
Clear lists of square numbers, cube numbers, and square roots, plus what the terms actually mean and the patterns that make them easy to remember.
What are square numbers?
A square number is what you get when you multiply a whole number by itself: 4 × 4 = 16, so 16 is a square number, written 4². The name is literal. Sixteen dots arrange into a perfect four-by-four square. Children in England meet square numbers in Year 5, and they keep appearing all the way through GCSE.
What are cube numbers?
A cube number multiplies a whole number by itself twice: 3 × 3 × 3 = 27, written 3³. Again the name is literal: 27 unit blocks stack into a perfect three-by-three-by-three cube. The first five, worth knowing cold, are 1, 8, 27, 64, 125.
Square numbers 1 to 20
| n | n² |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| 11 | 121 |
| 12 | 144 |
| 13 | 169 |
| 14 | 196 |
| 15 | 225 |
| 16 | 256 |
| 17 | 289 |
| 18 | 324 |
| 19 | 361 |
| 20 | 400 |
Cube numbers 1 to 20
| n | n³ |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1000 |
| 11 | 1331 |
| 12 | 1728 |
| 13 | 2197 |
| 14 | 2744 |
| 15 | 3375 |
| 16 | 4096 |
| 17 | 4913 |
| 18 | 5832 |
| 19 | 6859 |
| 20 | 8000 |
Square roots 1 to 20
| n | √n |
|---|---|
| 1 | 1 |
| 2 | 1.414 |
| 3 | 1.732 |
| 4 | 2 |
| 5 | 2.236 |
| 6 | 2.449 |
| 7 | 2.646 |
| 8 | 2.828 |
| 9 | 3 |
| 10 | 3.162 |
| 11 | 3.317 |
| 12 | 3.464 |
| 13 | 3.606 |
| 14 | 3.742 |
| 15 | 3.873 |
| 16 | 4 |
| 17 | 4.123 |
| 18 | 4.243 |
| 19 | 4.359 |
| 20 | 4.472 |
Tricks for remembering squares and cubes
- Squares grow by odd numbers. 1, 4, 9, 16, 25: the gaps are 3, 5, 7, 9. To get the next square, add the next odd number. This turns the whole list into a pattern instead of twenty separate facts.
- Squares of numbers ending in 5 always end in 25, and the front is the tens digit times one more than itself: 35² = (3 × 4) then 25, so 1225.
- Cube endings follow a fixed pattern. Cubes of numbers ending in 1, 4, 5, 6, or 9 end in the same digit. The pairs 2 and 8, and 3 and 7, swap endings. So 12³ must end in 8.
- 64 is the celebrity number. It is both a square (8²) and a cube (4³), the smallest such number after 1. Spotting it delights children and quietly teaches the idea of a sixth power.
Why quick recall of squares and cubes matters
Square and cube numbers are recall facts, the same species as times tables. A child who instantly knows 13² = 169 has spare working memory left for the actual problem, while a child who has to compute it from scratch spends their focus before the real question begins. That is the entire case for mental arithmetic training: automatic facts free the mind for thinking. We explain the research behind this in our guide to math fact fluency.
On an abacus, a square like 14 × 14 is worked one place value at a time, which is why abacus-trained children often find squares less scary: the big number is just a sequence of small, familiar moves. You can try it yourself on our free virtual abacus, and the full times tables that underpin every square and cube live on our times tables 1 to 20 chart.
Common questions
What are cube numbers?
A cube number is the result of multiplying a whole number by itself twice, written n cubed. For example, 2 cubed is 2 x 2 x 2 = 8. The first ten cube numbers are 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000. They are called cube numbers because that many unit blocks can be stacked into a perfect cube.
What are square numbers?
A square number is the result of multiplying a whole number by itself, written n squared. For example, 5 squared is 5 x 5 = 25. The first ten square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. They are called square numbers because that many dots can be arranged into a perfect square.
Which numbers are both square and cube numbers?
A number that is both a square and a cube is a sixth power. The smallest examples are 1 (1 squared and 1 cubed) and 64 (8 squared and 4 cubed). The next is 729, which is 27 squared and 9 cubed.
When do children learn square and cube numbers at school?
In England, square and cube numbers are introduced in upper Key Stage 2, typically Year 5 and Year 6, and square roots follow in Key Stage 3. Other curricula introduce them at a similar age, usually between ages 9 and 12.
Is there a trick for remembering cube numbers?
Learn the first five cold: 1, 8, 27, 64, 125. Then notice the endings repeat in a pattern: cubes of numbers ending in 1 end in 1, ending in 4 end in 4, ending in 5 end in 5, ending in 6 end in 6, and ending in 9 end in 9. Pairs swap for the rest: 2 and 8, 3 and 7. So 13 cubed must end in 7.
Turn number facts into number sense
Our live online abacus classes teach children to calculate mentally, so squares, cubes, and tables become numbers they can picture. Book a free 30-minute demo class.
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