30-60-90 e 45-45-90 — memorize uma vez, resolva instantaneamente
Revisado por [email protected], Geometry Calculator Developer & Online Math Educator Última atualização May 21, 2026
Dois triângulos retângulos aparecem com tanta frequência em geometria, trigonometria e física que as razões dos lados valem a pena memorizar: o triângulo 30-60-90 (lados 1 : √3 : 2) e o triângulo 45-45-90 (lados 1 : 1 : √2). Uma vez que você reconhece os ângulos, pode obter os lados sem calcular — economizando um cálculo de Pitágoras a cada vez.
| Nome | Fórmula | Notas |
|---|---|---|
| Razão dos lados 30-60-90 | 1 : √3 : 2 |
Opostos aos ângulos de 30°, 60°, 90° respectivamente. Cateto menor, cateto maior, hipotenusa. |
| Cateto maior 30-60-90 | long_leg = short_leg × √3 |
O lado oposto a 60° é √3 vezes o lado oposto a 30°. |
| Hipotenusa 30-60-90 | hypotenuse = 2 × short_leg |
A hipotenusa é sempre exatamente o dobro do lado mais curto (oposto a 30°). |
| Cateto menor a partir da hipotenusa 30-60-90 | short_leg = hypotenuse / 2 |
Inverso: se você conhece a hipotenusa, metade dela dá o lado oposto a 30°. |
| Cateto maior a partir da hipotenusa 30-60-90 | long_leg = (hypotenuse / 2) × √3 |
Substitua cateto_menor = hip/2 em cateto_maior = menor × √3. |
| Razão dos lados 45-45-90 | 1 : 1 : √2 |
Ambos os catetos são iguais (triângulo retângulo isósceles); a hipotenusa é cateto × √2. |
| Hipotenusa 45-45-90 | hypotenuse = leg × √2 |
Para qualquer triângulo retângulo com catetos iguais, a hipotenusa é √2 ≈ 1,414 vezes qualquer cateto. |
| Cateto a partir da hipotenusa 45-45-90 | leg = hypotenuse / √2 = hypotenuse·√2/2 |
Inverso — quando você conhece a hipotenusa, divida por √2. |
| 45-45-90 a partir do quadrado | half a square diagonally = 45-45-90 |
Um triângulo 45-45-90 é exatamente metade de um quadrado cortado ao longo de sua diagonal. É por isso que s² + s² = (s√2)² = 2s². |
| 30-60-90 a partir do equilátero | half an equilateral cut by altitude |
Um triângulo 30-60-90 é metade de um triângulo equilátero cortado por sua altura. É daí que vêm as razões 1 : √3 : 2. |
| sen(30°), cos(60°) | = 1/2 |
Leia diretamente do triângulo 30-60-90: oposto/hipotenusa = menor/2 = (1)/(2) quando menor = 1. |
| sen(60°), cos(30°) | = √3/2 |
Oposto/hipotenusa = maior/2 = √3/2 no triângulo 1:√3:2. |
| sen(45°), cos(45°) | = √2/2 ≈ 0.707 |
Em um 45-45-90, o oposto e o adjacente do ângulo são ambos catetos (ambos = 1), e a hipotenusa = √2. Então sen = 1/√2 = √2/2. |
| tg(30°) | = 1/√3 = √3/3 |
oposto/adjacente = menor/maior = 1/√3. |
| tg(45°) | = 1 |
oposto/adjacente = cateto/cateto = 1. |
| tg(60°) | = √3 |
oposto/adjacente = maior/menor = √3/1. |
Efficiently solving for unknown dimensions in special right triangles requires understanding the specific input parameters for each triangle type. Our calculator simplifies this process by allowing you to derive all missing sides and angles from a single known value. To begin, identify which type of triangle you are working with: either a 45-45-90 triangle or a 30-60-90 triangle. These shapes have fixed angle measures and consistent side ratios, which allows for precise calculations without needing complex trigonometric functions.
First, enter the known side length into the designated input field. This value can be either a leg or the hypotenuse, depending on the information provided in your problem. For a 45-45-90 triangle, the two legs are always equal in length. If you know the length of one leg, the other leg is identical. If you know the hypotenuse, you can find the leg length by dividing the hypotenuse by √2. Conversely, for a 30-60-90 triangle, the sides follow a ratio of 1 : √3 : 2. The shortest side is opposite the 30-degree angle, the longer leg is opposite the 60-degree angle, and the hypotenuse is twice the length of the shortest side.
After entering your value, select the corresponding triangle type from the dropdown menu. This ensures the calculator applies the correct geometric relationships. Finally, click the calculate button. The tool will instantly compute the remaining side lengths and confirm the angle measurements. While these calculators rely on fixed ratios, they are mathematically equivalent to using trigonometric ratios like sine, cosine, and tangent, but with significantly fewer steps. Additionally, you can verify your results manually using the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (a² + b² = c²). This step-by-step approach ensures accuracy and saves time when solving geometry problems involving these specific triangular forms.
Even with standardized shapes, errors frequently occur when applying geometric principles to 30-60-90 triangle and 45-45-90 triangle problems. One of the most prevalent issues involves confusing the roles of the legs and the hypotenuse. In a 45-45-90 triangle, students often mistakenly assume the hypotenuse is equal to the leg length multiplied by two. However, the correct relationship is that the hypotenuse is the leg length multiplied by √2. Similarly, in a 30-60-90 triangle, the hypotenuse is twice the length of the shorter leg, not the longer leg. Misidentifying which side corresponds to which part of the ratio leads to significant calculation errors.
Another common pitfall is misapplying side ratios across different triangle types. The ratio 1 : √3 : 2 is exclusive to the 30-60-90 configuration, while the 45-45-90 triangle follows the 1 : 1 : √2 pattern. Using the wrong ratio set invalidates the entire solution. It is crucial to recognize the angle measures first before selecting the appropriate mathematical model. Furthermore, many learners forget that the hypotenuse is always the longest side in any right triangle. This fundamental property serves as a quick check for validity; if a calculated hypotenuse is shorter than a leg, the result is physically impossible. Relying solely on memorized shortcuts without verifying against the Pythagorean theorem can also lead to mistakes. Understanding the underlying geometry rather than just the formula helps prevent these errors. By carefully distinguishing between the leg and the hypotenuse and respecting the unique properties of each triangle type, you can ensure accurate results in your geometric computations.
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