# Special Right Triangles Calculator

URL: https://speedycalculator.com/calculators/math/special-right-triangles/
Last updated: 2026-10-30

> The special right triangles calculator solves 45-45-90, 30-60-90, 1:2, 1:3 and 3-4-5 right triangles from the short leg, giving all sides, angles, area and perimeter.

## Inputs

- Triangle type
- Short leg a (cm, m, mm, in, ft, yd, km, mi)

## Results

- Long leg b
- Hypotenuse c
- Smallest angle α
- Angle β
- Area
- Perimeter

## Formula

```
45-45-90: 1 : 1 : √2 · 30-60-90: 1 : √3 : 2 · 3-4-5: 3 : 4 : 5
Legs 1 : 2 → hypotenuse √5 · legs 1 : 3 → hypotenuse √10
```

## Explanation

Special right triangles have side ratios you can remember, so many problems can be solved without a calculator. Builders use the 3-4-5 triangle to set out square corners; the 1:2 and 1:3 triangles appear in slopes and stairs.

### How to use the Special Right Triangles Calculator

- **Triangle type**
- **Short leg a**

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

### How it works

Choose the triangle and enter the short leg a. The other sides follow from the fixed ratio, and the angles from arctan.

### Inputs explained

FieldWhat to enterTriangle typeChoose one: 45° – 45° – 90°, 30° – 60° – 90°, Legs 1 : 2 (b = 2a), Legs 1 : 3 (b = 3a) …Short leg aUnits: cm, m, mm, in, ft, yd …

### What each result means

- **Hypotenuse c** = sqrt([Short leg a]^2 + [Long leg b]^2)
- **Smallest angle α** = deg(atan([Short leg a]÷[Long leg b]))
- **Angle β** = 90 - [Smallest angle α]
- **Area** = [Short leg a]×[Long leg b]÷2
- **Perimeter** = [Short leg a] + [Long leg b] + [Hypotenuse c]

### Example values

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

Short leg a (cm)Hypotenuse c (cm)1.532.254.5363.757.54.59612

## Worked example

Inputs — Triangle type: 30° – 60° – 90°; Short leg a: 3 cm.

Results — Long leg b = 5.196152 cm; **Hypotenuse c = 6 cm**; Smallest angle α = 30 °; Angle β = 60 °; Area = 7.794229 cm²; Perimeter = 14.196152 cm.

## Key terms

- **Ratio**: A fixed proportion between lengths.

## FAQ

**Why is 3-4-5 useful?**
It is the smallest right triangle with whole-number sides – perfect for setting out right angles.

**What are the two classic special triangles?**
45-45-90 and 30-60-90.

**Is any multiple of 3-4-5 a right triangle?**
Yes, e.g. 6-8-10 or 30-40-50.

**What angle does a 1:2 slope make?**
arctan(1/2) ≈ 26.57°.


## 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 – Special right triangle](https://mathworld.wolfram.com/SpecialRightTriangle.html)