# Triangle Inequality Calculator

URL: https://speedycalculator.com/calculators/math/triangle-inequality/
Last updated: 2026-10-30

> The triangle inequality calculator checks whether three lengths can form a triangle and, if not, which inequality is violated – and gives the range the third side must lie in.

## Inputs

- Side a (cm, m, mm, in, ft, yd, km, mi)
- Side b (cm, m, mm, in, ft, yd, km, mi)
- Side c (cm, m, mm, in, ft, yd, km, mi)

## Results

- Can these sides form a triangle?
- a + b > c
- a + c > b
- b + c > a
- Side c must be greater than
- Side c must be less than

## Formula

```
a + b > c,  a + c > b,  b + c > a  ⇔  |a − b| < c < a + b
```

## Explanation

Not every three lengths make a triangle: if one stick is longer than the other two together, they cannot meet. The triangle inequality is the rule, and it also says that a straight line is the shortest path between two points.

### How to use the Triangle Inequality Calculator

- **Side a**
- **Side b**
- **Side c**

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

### How it works

Enter three lengths. The calculator tests the three inequalities, reports any that fail and gives the allowed range for side c. Equality means a degenerate (flat) triangle.

### Inputs explained

FieldWhat to enterSide aUnits: cm, m, mm, in, ft, yd …Side bUnits: cm, m, mm, in, ft, yd …Side cUnits: cm, m, mm, in, ft, yd …

### What each result means

- **Side c must be greater than** = abs([Side a] - [Side b])
- **Side c must be less than** = [Side a] + [Side b]

## Worked example

Inputs — Side a: 4 cm; Side b: 6 cm; Side c: 9 cm.

Results — **Can these sides form a triangle? = Yes**; a + b > c = Holds; a + c > b = Holds; b + c > a = Holds; Side c must be greater than = 2 cm; Side c must be less than = 10 cm.

## Key terms

- **Degenerate triangle**: Three collinear points – a triangle with zero area.

## FAQ

**Can 2, 3 and 6 form a triangle?**
No – 2 + 3 < 6.

**What if a + b = c exactly?**
The "triangle" is flat (degenerate) with zero area.

**Why is it called an inequality?**
It is a "greater than" condition, not an equation.

**Does it hold in higher dimensions?**
Yes – for any distance: |x + y| ≤ |x| + |y|.


## Sources

- [OpenStax – Algebra and Trigonometry 2e](https://openstax.org/details/books/algebra-and-trigonometry-2e)
- [OpenStax – Prealgebra 2e](https://openstax.org/details/books/prealgebra-2e)
- [Wolfram MathWorld – Triangle inequality](https://mathworld.wolfram.com/TriangleInequality.html)