# Right triangle calculator: hypotenuse, sides and angles

> Hypotenuse, missing leg, both acute angles, area and perimeter of a right triangle from any two values, using Pythagoras' theorem and SOHCAHTOA.

Interactive version: https://www.calcopenly.com/geometry/right-triangle-calculator
Subject: Geometry calculators

A right triangle is fixed by any two of its parts as long as one of them is a side. With both legs known, Pythagoras' theorem gives the hypotenuse c = √(a² + b²). With a side and an acute angle, the rest follows from sin α = a/c, cos α = b/c and tan α = a/b, and the two acute angles always add up to 90°.

The default legs of 3 cm and 4 cm make the 3-4-5 triangle: hypotenuse 5 cm, area 6 cm², perimeter 12 cm and angles of 36.8699° and 53.1301°. Builders use that ratio to square a corner, since marks 3 m and 4 m from it along two walls should be exactly 5 m apart. The results also include the height onto the hypotenuse (ab/c), the inradius ((a + b − c)/2) and the circumradius (c/2).

Angle α is opposite leg a and β is opposite leg b. Angles can be typed in degrees, radians, gradians or turns and must lie strictly between 0° and 90°.

## Inputs

- **I know** (options: Both legs (a and b), Leg a and hypotenuse c, Leg b and hypotenuse c, Leg a and angle α, Leg b and angle α, Hypotenuse c and angle α, Leg a and angle β, Leg b and angle β, Hypotenuse c and angle β)
- **Leg a (opposite α)**
- **Leg b (opposite β)**
- **Hypotenuse c**
- **Angle α**: Between 0° and 90°
- **Angle β**: Between 0° and 90°
- **Show results in** (options: Millimetres (mm), Centimetres (cm), Metres (m), Kilometres (km), Inches (in), Feet (ft), Yards (yd), Miles (mi))

## Results

- Area — main result
- Leg a
- Leg b
- Hypotenuse c
- Angle α (°)
- Angle β (°)
- Perimeter
- Height onto the hypotenuse
- Inradius
- Circumradius

## Formula

$$
a^2 + b^2 = c^2,\quad \sin\alpha = \frac{a}{c},\quad \cos\alpha = \frac{b}{c},\quad \tan\alpha = \frac{a}{b},\quad \alpha + \beta = 90^\circ
$$

## Worked examples

### Legs 3 and 4

- I know: Both legs (a and b)
- Leg a (opposite α): 3 cm
- Leg b (opposite β): 4 cm
- Show results in: Centimetres (cm)
- **Hypotenuse c: 5 cm**
- **Area: 6 cm²**
- **Angle α: 36.869898 °**
- **Angle β: 53.130102 °**
- **Height onto the hypotenuse: 2.4 cm**
- **Inradius: 1 cm**
- **Circumradius: 2.5 cm**
- **Perimeter: 12 cm**
- Checked against: 3-4-5 triple; h = 3·4/5; r = (3 + 4 − 5)/2; Python 3.8 math: degrees(atan(3/4))

### Leg 5 and hypotenuse 13

- I know: Leg a and hypotenuse c
- Leg a (opposite α): 5 cm
- Hypotenuse c: 13 cm
- Show results in: Centimetres (cm)
- **Leg b: 12 cm**
- **Area: 30 cm²**
- **Perimeter: 30 cm**
- **Angle α: 22.619865 °**
- Checked against: 5-12-13 triple; Python 3.8 math: degrees(asin(5/13))

### 30-60-90 from hypotenuse 10 and α = 30°

- I know: Hypotenuse c and angle α
- Hypotenuse c: 10 cm
- Angle α: 30 °
- Show results in: Centimetres (cm)
- **Leg a: 5 cm**
- **Leg b: 8.660254 cm**
- **Angle β: 60 °**
- **Area: 21.650635 cm²**
- Checked against: Side opposite 30° is half the hypotenuse; Python 3.8 math: 10*cos(radians(30)), 25*sqrt(3)/2

### 45° angle gives equal legs (edge case)

- I know: Leg a and angle α
- Leg a (opposite α): 7 cm
- Angle α: 45 °
- Show results in: Centimetres (cm)
- **Leg b: 7 cm**
- **Hypotenuse c: 9.899495 cm**
- **Area: 24.5 cm²**
- **Angle β: 45 °**
- Checked against: tan 45° = 1 so b = a; Python 3.8 math: 7*sqrt(2)

### Leg b = 9 and β = 60°

- I know: Leg b and angle β
- Leg b (opposite β): 9 cm
- Angle β: 60 °
- Show results in: Centimetres (cm)
- **Leg a: 5.196152 cm**
- **Hypotenuse c: 10.392305 cm**
- **Angle α: 30 °**
- Checked against: Python 3.8 math: 9/tan(radians(60)), 9/sin(radians(60))

### Legs 6 in and 8 in, results in cm

- I know: Both legs (a and b)
- Leg a (opposite α): 6 in
- Leg b (opposite β): 8 in
- Show results in: Centimetres (cm)
- **Hypotenuse c: 25.4 cm**
- **Area: 154.8384 cm²**
- Checked against: 6-8-10 in triangle; 10 in = 25.4 cm and 24 in² = 24 × 6.4516 cm² exactly (1 in = 2.54 cm)

## Questions

### How do you find the hypotenuse of a right triangle?

Square both legs, add them and take the square root: c = √(a² + b²). Legs of 3 and 4 give √25 = 5, and legs of 6 in and 8 in give 10 in. If you know one leg and the angle opposite it instead, divide by the sine: c = a / sin α, so a 5 cm leg opposite 30° means a 10 cm hypotenuse.

### How do you find a missing side of a right triangle?

For a missing leg, subtract the squares: b = √(c² − a²). A hypotenuse of 13 and a leg of 5 give √(169 − 25) = 12. The hypotenuse must be longer than either leg, or no right triangle exists. With one side and an angle, use SOHCAHTOA: opposite = hypotenuse × sin α and adjacent = hypotenuse × cos α.

### What are the side ratios of a 30-60-90 and a 45-45-90 triangle?

A 30-60-90 triangle has sides in the ratio 1 : √3 : 2, so the side opposite 30° is half the hypotenuse; a hypotenuse of 10 gives legs of 5 and 8.6603. A 45-45-90 triangle has equal legs and a hypotenuse √2 ≈ 1.4142 times a leg, so legs of 7 give a hypotenuse of 9.8995.

### What does SOHCAHTOA mean?

It is a memory aid for the three trigonometric ratios in a right triangle: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent. In the 3-4-5 triangle, the angle opposite the side of 3 has sin = 0.6, cos = 0.8 and tan = 0.75, and each inverse function returns the same 36.87°.

### How do you check a corner is square with the 3-4-5 rule?

Mark 3 units along one side and 4 along the other, measured from the corner; the diagonal between the marks is exactly 5 when the angle is 90°. A longer diagonal means the angle is too wide: 5.05 m on 3 m and 4 m legs is 91.2°, and 4.95 m is 88.8°. Multiples such as 6-8-10 or 9-12-15 give a more precise check on large slabs.

### How accurate is the right triangle calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 6 worked examples whose answers come from independent sources; for example, “Legs 3 and 4” is checked against 3-4-5 triple; h = 3·4/5; r = (3 + 4 − 5)/2; Python 3.8 math: degrees(atan(3/4)).

### Where does the method come from?

Euclid, Elements, Book I, Proposition 47 (Pythagorean theorem); OpenStax Precalculus 2e, §5.4 Right Triangle Trigonometry.

## Sources

- [Euclid, Elements, Book I, Proposition 47 (Pythagorean theorem)](https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI47.html)
- [OpenStax Precalculus 2e, §5.4 Right Triangle Trigonometry](https://openstax.org/books/precalculus-2e/pages/5-4-right-triangle-trigonometry)