# How do I solve special right triangles?

Solves a 30-60-90 or 45-45-90 triangle from one side, 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/special-right-triangles-calculator
- JSON spec: https://www.acalculator.org/math/special-right-triangles-calculator.json
- Version: 4ffae590ce9a

## Default answer

Example with the default inputs (Triangle 30-60-90, You know the Short leg a, Its value 5): A 30°, 60°, 90° triangle with sides 5, 8.660254 and 10 has an area of 21.650635.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| type | Triangle | The 30-60-90 triangle (half an equilateral triangle) or the 45-45-90 triangle (half a square). |
| know | You know the | Which part of the triangle you know. |
| know45 | 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

30-60-90: a : b : c = 1 : √3 : 2, area = a²√3 ÷ 2, perimeter = a(3 + √3). 45-45-90: a : b : c = 1 : 1 : √2, area = a² ÷ 2, perimeter = a(2 + √2). Height to the hypotenuse = a × b ÷ c.

## Assumptions

- a is the side opposite the smallest angle (30° or 45°), b the other leg, 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. type = 30-60-90, 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; 45-45-90 sides s, s, √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. type = 45-45-90, know45 = leg, v = 5 gives a = 5, b = 5, c = 7.071068, area = 12.5, perimeter = 17.071068, exact = a = 5, b = 5, c = 5√2, angles = 45°, 45°, 90°. Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s; 45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry.
3. type = 30-60-90, know = long, v = 6 gives a = 3.464102, c = 6.928203, area = 10.392305, 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; 45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry.
4. type = 45-45-90, know45 = area, v = 50 gives a = 10, b = 10, c = 14.142136, perimeter = 34.142136. Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (30-60-90 sides s, √3s, 2s; 45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry.
5. type = 30-60-90, 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; 45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry.

## FAQ

### What are the special right triangles?

The 30-60-90 triangle and the 45-45-90 triangle. Their sides are always in the same ratio: 1 : √3 : 2 for the 30-60-90 triangle and 1 : 1 : √2 for the 45-45-90 triangle, so one side gives the other two.

### Where do the ratios come from?

A 30-60-90 triangle is half of an equilateral triangle, so the hypotenuse is twice the short leg, and the Pythagorean theorem gives the long leg: √(2² − 1²) = √3. A 45-45-90 triangle is half of a square, so its legs are equal and the hypotenuse is √(1² + 1²) = √2.

### How do I find the sides of a 30-60-90 triangle from the hypotenuse?

Halve the hypotenuse to get the short leg, then multiply the short leg by √3 for the long leg. A hypotenuse of 10 gives a short leg of 5 and a long leg of 5√3 ≈ 8.66.

### How do I find the hypotenuse of a 45-45-90 triangle?

Multiply a leg by √2. Legs of 5 give a hypotenuse of 5√2 ≈ 7.071. To go back, divide the hypotenuse by √2: 10 ÷ √2 = 5√2 ≈ 7.071.

### Why is the long leg divided by √3 written as a multiple of √3?

Dividing by √3 is the same as multiplying by √3 ÷ 3, which removes the root from the bottom. A long leg of 6 gives a short leg of 6 ÷ √3 = 2√3.

### Can I start from the area or the perimeter?

Yes. For a 45-45-90 triangle the area is a² ÷ 2, so an area of 50 gives legs of 10. For a 30-60-90 triangle the perimeter is a(3 + √3), so a perimeter of 30 gives a short leg of 30 ÷ (3 + √3) ≈ 6.34.

## Sources

- OpenStax, Algebra and Trigonometry 2e, section 7.2 Right Triangle Trigonometry: the 30-60-90 triangle has sides s, √3s, 2s and the 45-45-90 triangle sides s, s, √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