Two triangles make a square
Add two neighbouring triangular numbers.
Triangles of dots
1, 3, 6, 10, 15, … are triangular numbers: dots arranged in a triangle. Add two neighbours: 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16.
What will 10 + 15 be?
What this lesson covers
The idea
The sum of two consecutive triangular numbers (1, 3, 6, 10, 15, …) is a perfect square.
Triangles of dots
1, 3, 6, 10, 15, … are triangular numbers: dots arranged in a triangle. Add two neighbours: 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16.
What will 10 + 15 be?
- 24
- 25
- 26
Join the triangles
Pick a square size, then press Join them. How do the two triangles fit?
Two triangles, one square
The sum of two consecutive triangular numbers is a perfect square.
The two triangles fit together like two halves of a square of dots.
- Neighbours | Sum | Square
- 1 + 3 | 4 | 2²
- 3 + 6 | 9 | 3²
- 6 + 10 | 16 | 4²
- 10 + 15 | 25 | 5²
- 15 + 21 | 36 | 6²
Notes
The sum of two consecutive triangular numbers is a perfect square.
Check yourself
The triangular numbers go 1, 3, 6, 10, 15, 21, 28, … What is 21 + 28?
Answer: 49
21 + 28 = 49 = 7².
Which two neighbouring triangular numbers add up to 64?
A 10 × 10 square of dots is split into two triangles. One triangle has 55 dots. How many dots does the other have?
Answer: 45
100 − 55 = 45. And 45 + 55 = 100 = 10².
- 21 + 28. 21 + 28 = 49, not 64.
- 28 + 36 — correct. Yes! 28 + 36 = 64 = 8².
- 36 + 45. 36 + 45 = 81, not 64.
- 15 + 21. 15 + 21 = 36, not 64.