TablesWhat squares and cubes are
The square of 7 is 7 × 7 = 49. It is called a square because 49 small tiles make a square floor, 7 tiles long and 7 wide. The cube of 7 is 7 × 7 × 7 = 343, the number of small blocks in a cube that is 7 blocks long, wide and high. Squares are written with a small 2, as in 7², and cubes with a small 3, as in 7³.
Patterns in the squares
- Last digits. A square can end only in 0, 1, 4, 5, 6 or 9. No square ends in 2, 3, 7 or 8. So 247 cannot be a square.
- Odd-number steps. From one square to the next you add the odd numbers in turn: 1, then + 3 = 4, + 5 = 9, + 7 = 16, + 9 = 25. So if you know 20² = 400, then 21² = 400 + 20 + 21 = 441.
- Pairs that end alike. 4² = 16 and 6² = 36 both end in 6. 3² = 9 and 7² = 49 both end in 9. Numbers that add up to 10 have squares with the same last digit.
- Even and odd. The square of an even number is even, and of an odd number is odd.
Quick ways to square
Numbers ending in 5. Multiply the tens digit by the next number up and write 25 after it. For 25²: 2 × 3 = 6, so 625. For 65²: 6 × 7 = 42, so 4225.
Numbers near a round number. For 29², start from 30² = 900 and step back: 900 − 30 − 29 = 841. For 31²: 900 + 30 + 31 = 961.
Splitting. 23² is 20² plus twice 20 × 3 plus 3²: 400 + 120 + 9 = 529. This is the rule (a + b)² = a² + 2ab + b², met in class 8.
Patterns in the cubes
- The cubes of 1 to 9 end in 1, 8, 7, 4, 5, 6, 3, 2, 9. Each last digit appears once, so the last digit of a perfect cube tells you the last digit of its cube root.
- Add the cubes from 1 upwards and you always get a square: 1 + 8 = 9, 1 + 8 + 27 = 36, 1 + 8 + 27 + 64 = 100. These are the squares of 3, 6 and 10.
- A cube can be negative, a square cannot: (−2)³ = −8 but (−2)² = 4.
Which to learn by heart
Squares to 20 are met constantly from class 6 onwards, in area, Pythagoras and square roots. Squares to 30 and cubes to 15 are a common target for competitive exams. Learn them five at a time, and use the odd-number steps above to rebuild any that slip.
Squares sit on the diagonal of the multiplication chart. For area and volume formulas that use squares and cubes, see our neighbour site Shapes.