Square and triangular numbers: building shapes from blocks
Square numbers
A square number is what you get when you build a square from blocks: 1, 4, 9, 16, 25 and so on. The fifth square has 5 rows of 5, so 5 × 5 = 25.
To get from one square to the next you add an L-shaped border. From 4 to 9 you add 5 blocks, from 9 to 16 you add 7, from 16 to 25 you add 9. The borders are the odd numbers, so adding the first n odd numbers always gives the nth square.
Triangular numbers
Stack rows of 1, 2, 3, 4 blocks to make a staircase. The totals are 1, 3, 6, 10, 15, 21: the triangular numbers. Each one adds a row that is one longer than the last.
Two triangles make a rectangle
Build the staircase for 4 (10 blocks), then build a second copy and turn it upside down. Pushed together they make a 4 × 5 rectangle of 20 blocks. So one staircase is half of 4 × 5. In general the nth triangular number is n × (n + 1) ÷ 2. This is a shortcut for adding 1 + 2 + 3 + … + 100: it is 100 × 101 ÷ 2 = 5,050.
| n | Square n × n | Triangular n(n+1)/2 |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 3 |
| 3 | 9 | 6 |
| 4 | 16 | 10 |
| 5 | 25 | 15 |
| 6 | 36 | 21 |
| 8 | 64 | 36 |
Notice 36 appears in both columns. It is the square of 6 and the 8th triangular number.
The Number Blocks Tool flags square and triangular numbers. Try 36, 45 and 49. Open the tool with 36
Check yourself
Answer each question. Drawing the blocks can help.
- What is the 7th square number?
- What is the 7th triangular number?
- Add any two triangular numbers that sit next to each other, for example 6 + 10. What kind of number do you get?
Show answers
- 49
- 28
- A square number: 6 + 10 = 16, 10 + 15 = 25.