# How do I solve a 45-45-90 triangle?

Solves a 45-45-90 (isosceles right) triangle from a leg, the hypotenuse, the area or the perimeter: the sides with exact square roots, the area, the perimeter and the height.

- Page: https://www.acalculator.org/math/45-45-90-triangle-calculator
- JSON spec: https://www.acalculator.org/math/45-45-90-triangle-calculator.json
- Version: 5251626c7307

## Default answer

Example with the default inputs (You know the Leg a, Its value 5): A 45-45-90 triangle with legs 5 has a hypotenuse of 7.071068 and an area of 12.5.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| 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 | Leg a | The side opposite the smallest angle (30° or 45°). |
| b | 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 : 1 : √2. Area = a² ÷ 2, perimeter = a(2 + √2), height to the hypotenuse = c ÷ 2.

## Assumptions

- a and b are the equal legs, 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. know45 = leg, v = 5 gives a = 5, b = 5, c = 7.071068, area = 12.5, perimeter = 17.071068, height = 3.535534, exact = a = 5, b = 5, c = 5√2. Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry.
2. know45 = hyp, v = 10 gives a = 7.071068, b = 7.071068, area = 25, perimeter = 24.142136, exact = a = 5√2, b = 5√2, c = 10. Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry.
3. know45 = area, v = 50 gives a = 10, c = 14.142136, perimeter = 34.142136, height = 7.071068. Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry.
4. know45 = leg, v = 1 gives c = 1.414214, area = 0.5, angles = 45°, 45°, 90°, ratio = 1 : 1 : √2. Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (45-45-90 sides s, s, √2s), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry: the 45-45-90 triangle with legs 1 has hypotenuse √2.

## FAQ

### What is the 45-45-90 triangle rule?

The two legs are equal and the hypotenuse is a leg times √2, so the sides are in the ratio 1 : 1 : √2. It is half of a square, cut along the diagonal.

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

Multiply the leg by √2. Legs of 5 give a hypotenuse of 5√2 ≈ 7.071.

### How do I find the legs from the hypotenuse?

Divide the hypotenuse by √2, which is the same as multiplying by √2 ÷ 2. A hypotenuse of 10 gives legs of 5√2 ≈ 7.071.

### What is the area of a 45-45-90 triangle?

Half the square of a leg: a² ÷ 2. Legs of 5 give 12.5. From the area back, a = √(2 × area), so an area of 50 gives legs of 10.

### Is a 45-45-90 triangle isosceles?

Yes. It is the isosceles right triangle: two equal legs meet at the 90° angle, and the two other angles are 45° each.

### What is the height to the hypotenuse?

Half the hypotenuse: a × a ÷ (a√2) = a ÷ √2 = c ÷ 2. Legs of 1 give a hypotenuse of √2 ≈ 1.414 and a height of about 0.7071.

## Sources

- OpenStax, Algebra and Trigonometry 2e, section 7.2 Right Triangle Trigonometry: the 45-45-90 triangle has sides s, s, √2s. CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry