Square numbers
Theory
A square number is a whole number times itself, such as \(5 \times 5 = 25\). That many dots can be arranged in a perfect square. This Year 6 maths page shows how to find square numbers, work backwards to the number that was squared, and use the pattern of odd gaps between them.
A square number is the result of multiplying a whole number by itself. The square of \(13\) is \(13 \times 13 = 169\).
Square numbers can be shown as square arrays: \(n\) rows of \(n\) dots.
To check if a number is square, look for a whole number that multiplies by itself to give it. \(150\) is not square: \(12 \times 12 = 144\) and \(13 \times 13 = 169\).
To go from one square to the next, add an L-shaped border. The gaps between consecutive squares are the odd numbers \(3, 5, 7, 9, \ldots\)
Square numbers to know:
| n | \(n \times n\) | n | \(n \times n\) |
|---|---|---|---|
| 1 | 1 | 11 | 121 |
| 2 | 4 | 12 | 144 |
| 3 | 9 | 13 | 169 |
| 4 | 16 | 14 | 196 |
| 5 | 25 | 15 | 225 |
| 6 | 36 | 16 | 256 |
| 7 | 49 | 17 | 289 |
| 8 | 64 | 18 | 324 |
| 9 | 81 | 19 | 361 |
| 10 | 100 | 20 | 400 |
Square or not?
- Find the two square numbers either side.
- If the number equals one of them, it is square; otherwise it is not.
Next square from the gap
- Gap after \(n \times n\) is \(n + n + 1\).
- Add the gap to get the next square.
| \(15 \times 15\) | \(=\) | \(225\) |
| \(20 \times 20\) | \(=\) | \(400\) |
\(225\) and \(400\).
| \(17 \times 17\) | \(=\) | \(289\) |
\(17\).
| \(12 \times 12\) | \(=\) | \(144\) |
| \(13 \times 13\) | \(=\) | \(169\) |
No. It lies between \(144\) and \(169\).
| \(196 - 169\) | \(=\) | \(27\) |
| \(13 + 13 + 1\) | \(=\) | \(27\) |
\(27\), the odd number \(13 + 13 + 1\).
| \(49 - 36\) | \(=\) | \(13\) |
\(13\) squares.
| \(14 \times 14\) | \(=\) | \(196\) |
\(196\) tiles.
| \(3 \times 3\) | \(=\) | \(9\) |
| \(30 \times 30\) | \(=\) | \(900\) |
\(900\) (\(3 \times 3 = 9\), then two zeros).
| \(15 \times 15\) | \(=\) | \(225\) |
She doubled \(15\) instead of multiplying it by itself. The square is \(225\).
Common pitfalls
Frequently asked questions
What is a square number?
A whole number times itself, like \(13 \times 13 = 169\).
Why are they called square numbers?
That many dots can be arranged in a square.
How do I know if a number is square?
Find a whole number that multiplies by itself to give it.
What are the gaps between square numbers?
The odd numbers \(3, 5, 7, 9, \ldots\)
What is the square of 20?
\(400\).