# 30 60 90 Triangle Calculator

URL: https://speedycalculator.com/calculators/math/triangle-30-60-90/
Last updated: 2026-10-30

> The 30 60 90 triangle calculator finds every side, the area and the perimeter of a 30°-60°-90° right triangle from any one side – the short leg, the long leg or the hypotenuse.

## Inputs

- Short leg a (opposite 30°) (cm, m, mm, in, ft, yd, km, mi)

## Results

- Long leg b (opposite 60°)
- Hypotenuse c
- Area
- Perimeter
- Height to the hypotenuse

## Formula

```
a : b : c = 1 : √3 : 2
b = a√3,  c = 2a,  Area = a²√3/2
```

## Explanation

A 30-60-90 triangle is half of an equilateral triangle, so its sides are always in the ratio 1 : √3 : 2. It is one of the two "special" right triangles whose trigonometric values (sin 30° = 1/2, cos 30° = √3/2) every student learns by heart.

### How to use the 30 60 90 Triangle Calculator

- **Short leg a (opposite 30°)**
- Every result field is editable: type into **Long leg b (opposite 60°)** or any other result and **Short leg a (opposite 30°)** plus every other field update to match.

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

### How it works

Enter the short leg a – or type the long leg, hypotenuse, area or perimeter into its field and the short leg is worked out backwards.

### What each result means

- **Long leg b (opposite 60°)** = [Short leg a (opposite 30°)]×sqrt(3)
- **Hypotenuse c** = 2×[Short leg a (opposite 30°)]
- **Area** = [Short leg a (opposite 30°)]^2×sqrt(3)÷2
- **Perimeter** = [Short leg a (opposite 30°)]×(3 + sqrt(3))
- **Height to the hypotenuse** = [Short leg a (opposite 30°)]×sqrt(3)÷2

### Example values

How the main result changes when **Short leg a** changes and everything else stays at the example values above:

Short leg a (opposite 30°) (cm)Long leg b (opposite 60°) (cm)2.54.33013.756.495258.66036.2510.82537.512.99041017.3205

## Worked example

Inputs — Short leg a (opposite 30°): 5 cm.

Results — **Long leg b (opposite 60°) = 8.660254 cm**; Hypotenuse c = 10 cm; Area = 21.650635 cm²; Perimeter = 23.660254 cm; Height to the hypotenuse = 4.330127 cm.

## Key terms

- **Special right triangle**: A right triangle with simple exact side ratios.

## FAQ

**What are the side ratios of a 30-60-90 triangle?**
1 : √3 : 2.

**Which side is opposite 30°?**
The shortest leg, half of the hypotenuse.

**Why is it called special?**
Its sides and trig values have exact, easy-to-remember forms.

**How is it related to an equilateral triangle?**
Cutting an equilateral triangle by its altitude gives two 30-60-90 triangles.


## 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 – 30-60-90 triangle](https://mathworld.wolfram.com/30-60-90Triangle.html)