Triangular numbers
Theory
A triangular number is the number of dots in a triangle with rows of \(1, 2, 3, \ldots\) dots: \(1, 3, 6, 10, 15, \ldots\). Each one is a sum \(1 + 2 + 3 + \ldots\). This Year 6 maths page shows how to build and continue the pattern, why the gaps grow by one, and how triangular numbers appear in stacking and handshake problems.
A triangular number is \(1 + 2 + 3 + \ldots\) up to a counting number: a triangle of dots with that many rows. The position number tells you how many rows.
To get the next triangular number, add a new bottom row with one more dot than the last row. So the gaps are \(2, 3, 4, 5, \ldots\)
A number is triangular if it appears in the list. \(70\) is not: the list goes \(\ldots, 66, 78, \ldots\)
The first twelve triangular numbers:
| Position | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Triangular | 1 | 3 | 6 | 10 | 15 | 21 |
| Position | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|
| Triangular | 28 | 36 | 45 | 55 | 66 | 78 |
Finding a triangular number
- Add \(1 + 2 + 3 + \ldots\) up to the position number, or
- Start from a known triangular number and keep adding the next counting number.
| \(55 + 11\) | \(=\) | \(66\) |
| \(66 + 12\) | \(=\) | \(78\) |
\(66\) and \(78\).
No. The triangular numbers jump from \(66\) to \(78\).
| \(78 + 13\) | \(=\) | \(91\) |
\(91\).
| \(78 + 13 + 14\) | \(=\) | \(105\) |
\(1 + 2 + \ldots + 12 = 78\) (the \(12\)th), then add \(13\) and \(14\): \(105\), the \(14\)th triangular number.
That is the \(13\)th triangular number: \(91\) cans.
| \(11 + 10 + 9 + \ldots + 1\) | \(=\) | \(66\) |
The first person shakes \(11\) hands, the next \(10\) new hands, and so on: \(66\) handshakes (the \(11\)th triangular number).
In the Key facts table, \(66\) is the \(11\)th and the next gap is \(12\): \(66 + 12 = 78\), so \(78\) is the \(12\)th.
The gaps grow by one: \(55\) is the \(10\)th, so the next gap is \(11\) and the next number is \(66\).
Common pitfalls
Frequently asked questions
What is a triangular number?
A number of dots that can make a triangle: \(1, 3, 6, 10, 15, \ldots\)
How do you find the next triangular number?
Add the next counting number to the last one.
Why do the gaps grow by one?
Each new triangle adds a bottom row with one more dot than before.
How do I find a bigger one?
Start from one you know and keep adding the next counting number: the \(12\)th is \(78\), so the \(13\)th is \(91\) and the \(14\)th is \(105\).
Where do triangular numbers appear?
In stacks of cans or logs, bowling pins and handshake problems.