# Triangular Numbers Calculator

URL: https://speedycalculator.com/calculators/math/triangular-numbers/
Last updated: 2026-10-30

> The triangular numbers calculator gives the n-th triangular number Tₙ = n(n + 1)/2 and checks whether any number is triangular.

## Inputs

- Number n
- Is this number triangular?

## Results

- Triangular number Tₙ
- Next triangular number Tₙ₊₁
- Is the second number triangular?
- Its index k (Tₖ = number)

## Formula

```
Tₙ = 1 + 2 + … + n = n(n + 1) / 2
x is triangular ⇔ (√(8x + 1) − 1)/2 is a whole number
```

## Explanation

Triangular numbers count dots arranged in triangles: 1, 3, 6, 10, 15 … They count handshakes in a group (T₍ₙ₋₁₎), bowling pins (T₄ = 10) and appear in Pascal's triangle.

### How to use the Triangular Numbers Calculator

- **Number n**
- **Is this number triangular?**

Results update as you type, and everything is calculated in your browser.

### How it works

Gauss's trick pairs the numbers 1 … n from both ends to get n(n + 1)/2. The inverse formula tells you whether a number is triangular and, if so, which one.

### Inputs explained

FieldWhat to enterNumber nA numberIs this number triangular?A number

### What each result means

- **Triangular number Tₙ** = [Number n]×([Number n] + 1)÷2
- **Next triangular number Tₙ₊₁** = ([Number n] + 1)×([Number n] + 2)÷2
- **Its index k (Tₖ = number)** = (sqrt(8×[Is this number triangular?] + 1) - 1)÷2

### Example values

How the main result changes when **Number n** changes and everything else stays at the example values above:

Number nTriangular number Tₙ6219451278151201817124300

## Worked example

Inputs — Number n: 12; Is this number triangular?: 78.

Results — **Triangular number Tₙ = 78**; Next triangular number Tₙ₊₁ = 91; Is the second number triangular? = Yes; Its index k (Tₖ = number) = 12.

## Key terms

- **Triangular number**: A number of dots that can form an equilateral triangle.

## FAQ

**What are the first triangular numbers?**
1, 3, 6, 10, 15, 21, 28, 36, 45, 55.

**How many handshakes happen among 10 people?**
T₉ = 45.

**What do two consecutive triangular numbers add to?**
A perfect square: Tₙ + Tₙ₋₁ = n².

**Is 78 triangular?**
Yes, T₁₂ = 78.


## Sources

- [Wolfram MathWorld – Triangular number](https://mathworld.wolfram.com/TriangularNumber.html)
- [OpenStax – Contemporary Mathematics](https://openstax.org/details/books/contemporary-mathematics)