# Triângulo retângulo: hipotenusa, lados e ângulos

> Calcule hipotenusa, cateto desconhecido, ângulos agudos, área e perímetro com dois valores, usando o teorema de Pitágoras e trigonometria.

Versão interativa: https://www.calcopenly.com/pt/geometry/right-triangle-calculator
Tema: Calculadoras de geometria

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°.

## Dados

- **I know** (opções: 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°
- **Unidade dos resultados** (opções: Milímetros (mm), Centímetros (cm), Metros (m), Quilômetros (km), Polegadas (in), Pés (ft), Jardas (yd), Milhas (mi))

## Resultados

- Área — resultado principal
- Leg a
- Leg b
- Hypotenuse c
- Angle α (°)
- Angle β (°)
- Perímetro
- Height onto the hypotenuse
- Inradius
- Circumradius

## Fórmula

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

## Exemplos resolvidos

### Legs 3 and 4

- I know: Both legs (a and b)
- Leg a (opposite α): 3 cm
- Leg b (opposite β): 4 cm
- Unidade dos resultados: Centímetros (cm)
- **Hypotenuse c: 5 cm**
- **Área: 6 cm²**
- **Angle α: 36.869898 °**
- **Angle β: 53.130102 °**
- **Height onto the hypotenuse: 2.4 cm**
- **Inradius: 1 cm**
- **Circumradius: 2.5 cm**
- **Perímetro: 12 cm**
- Fonte de verificação: 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
- Unidade dos resultados: Centímetros (cm)
- **Leg b: 12 cm**
- **Área: 30 cm²**
- **Perímetro: 30 cm**
- **Angle α: 22.619865 °**
- Fonte de verificação: 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 °
- Unidade dos resultados: Centímetros (cm)
- **Leg a: 5 cm**
- **Leg b: 8.660254 cm**
- **Angle β: 60 °**
- **Área: 21.650635 cm²**
- Fonte de verificação: 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 °
- Unidade dos resultados: Centímetros (cm)
- **Leg b: 7 cm**
- **Hypotenuse c: 9.899495 cm**
- **Área: 24.5 cm²**
- **Angle β: 45 °**
- Fonte de verificação: 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 °
- Unidade dos resultados: Centímetros (cm)
- **Leg a: 5.196152 cm**
- **Hypotenuse c: 10.392305 cm**
- **Angle α: 30 °**
- Fonte de verificação: 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
- Unidade dos resultados: Centímetros (cm)
- **Hypotenuse c: 25.4 cm**
- **Área: 154.8384 cm²**
- Fonte de verificação: 6-8-10 in triangle; 10 in = 25.4 cm and 24 in² = 24 × 6.4516 cm² exactly (1 in = 2.54 cm)

## Perguntas

### 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.

### Qual é a precisão de “Triângulo retângulo: hipotenusa, lados e ângulos”?

A precisão depende dos dados inseridos e das hipóteses do método. O cálculo decimal usa 50 algarismos significativos, mas estimativas, métodos numéricos e dados de origem podem ter menor precisão; o arredondamento exibido não elimina essas limitações. Exemplos resolvidos verificados com fontes independentes: 6. Por exemplo, “Legs 3 and 4” é verificado com 3-4-5 triple; h = 3·4/5; r = (3 + 4 − 5)/2; Python 3.8 math: degrees(atan(3/4)).

### De onde vem o método?

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

## Fontes

- [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)