# How do I solve a 30-60-90 triangle?

Solves a 30-60-90 triangle from the short leg, the long leg, the hypotenuse, the area or the perimeter: all three sides with exact square roots, the area, the perimeter and the height.

- Page: https://www.acalculator.org/math/30-60-90-triangle-calculator
- JSON spec: https://www.acalculator.org/math/30-60-90-triangle-calculator.json
- Version: 2760fa5a62ea

## Default answer

Example with the default inputs (You know the Short leg a, Its value 5): A 30-60-90 triangle with short leg 5 has a long leg of 8.660254 and a hypotenuse of 10.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| know | You know the | Which part of the triangle you know. |
| v | Its value | The length of the known side, or the area or perimeter, in any one unit. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| c | Hypotenuse c | The longest side, opposite the 90° angle. |
| a | Short leg a | The side opposite the smallest angle (30° or 45°). |
| b | Long leg b | The other leg: a × √3 in a 30-60-90 triangle, equal to a in a 45-45-90 triangle. |
| area | Area | Half the product of the legs: a × b ÷ 2. |
| perimeter | Perimeter | The three sides added: a + b + c. |
| height | Height to the hypotenuse | The distance from the right angle to the hypotenuse: a × b ÷ c. |
| exact | Exact sides | The sides with square roots, when a side is known as a decimal with up to 6 places. |
| angles | Angles | The three angles of the triangle. |
| ratio | Side ratio a : b : c | The ratio of the three sides. |

## Method

a : b : c = 1 : √3 : 2. Area = a²√3 ÷ 2, perimeter = a(3 + √3), height to the hypotenuse = a√3 ÷ 2.

## Assumptions

- a is the short leg (opposite 30°), b the long leg (opposite 60°), c the hypotenuse.
- The known value is from 0.000001 to 1,000,000,000, in any one unit; the area is in that unit squared.
- Exact sides show when a side is known as a decimal with at most 6 places.

## Worked examples

1. know = short, v = 7 gives a = 7, b = 12.124356, c = 14, area = 42.435245, perimeter = 33.124356, height = 6.062178, exact = a = 7, b = 7√3, c = 14. Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry, Example 5: the side opposite 30° is 7, so the hypotenuse is 14 and the adjacent side 7√3 ≈ 12.1.
2. know = hyp, v = 10 gives a = 5, b = 8.660254, area = 21.650635, perimeter = 23.660254, exact = a = 5, b = 5√3, c = 10, ratio = 1 : √3 : 2. Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry.
3. know = long, v = 6 gives a = 3.464102, c = 6.928203, height = 3, exact = a = 2√3, b = 6, c = 4√3. Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry.
4. know = perimeter, v = 30 gives a = 6.339746, b = 10.980762, c = 12.679492, area = 34.807621. Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry.

## FAQ

### What is the 30-60-90 triangle rule?

The sides are always in the ratio 1 : √3 : 2. The short leg is opposite 30°, the long leg (short leg × √3) is opposite 60°, and the hypotenuse (short leg × 2) is opposite 90°.

### How do I find the long leg from the short leg?

Multiply by √3. A short leg of 7 gives a long leg of 7√3 ≈ 12.12 and a hypotenuse of 14.

### How do I find the short leg from the hypotenuse?

Halve it. A hypotenuse of 10 gives a short leg of 5 and a long leg of 5√3 ≈ 8.66.

### How do I find the short leg from the long leg?

Divide by √3, which is the same as multiplying by √3 ÷ 3. A long leg of 6 gives a short leg of 2√3 ≈ 3.464 and a hypotenuse of 4√3 ≈ 6.928.

### Why are the sides 1, √3 and 2?

A 30-60-90 triangle is half of an equilateral triangle with sides 2. Cutting it in half gives a short leg of 1 and a hypotenuse of 2, and the Pythagorean theorem gives the third side: √(4 − 1) = √3.

### What is the area of a 30-60-90 triangle?

Half the product of the legs: a × a√3 ÷ 2 = a²√3 ÷ 2. With a short leg of 7 that is 49√3 ÷ 2 ≈ 42.44.

## Sources

- OpenStax, Algebra and Trigonometry 2e, section 7.2 Right Triangle Trigonometry: the 30-60-90 triangle has sides s, √3s, 2s; Example 5 (side opposite 30° is 7, hypotenuse 14). CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry