30-60-90 y 45-45-90 — memoriza una vez, resuelve al instante
Revisado por [email protected], Geometry Calculator Developer & Online Math Educator Última actualización May 21, 2026
Dos triángulos rectángulos aparecen tan a menudo en geometría, trigonometría y física que vale la pena memorizar las razones de sus lados: el triángulo 30-60-90 (lados 1 : √3 : 2) y el triángulo 45-45-90 (lados 1 : 1 : √2). Una vez que reconoces los ángulos, puedes leer los lados sin calcular — ahorrando un cálculo de Pitágoras cada vez.
| Nombre | Fórmula | Notas |
|---|---|---|
| Razón de lados 30-60-90 | 1 : √3 : 2 |
Opuestos a los ángulos de 30°, 60°, 90° respectivamente. Cateto corto, cateto largo, hipotenusa. |
| Cateto largo 30-60-90 | long_leg = short_leg × √3 |
El lado opuesto a 60° es √3 veces el lado opuesto a 30°. |
| Hipotenusa 30-60-90 | hypotenuse = 2 × short_leg |
La hipotenusa siempre es exactamente el doble del lado más corto (opuesto a 30°). |
| Cateto corto 30-60-90 desde la hipotenusa | short_leg = hypotenuse / 2 |
Inverso: si conoces la hipotenusa, su mitad da el lado opuesto a 30°. |
| Cateto largo 30-60-90 desde la hipotenusa | long_leg = (hypotenuse / 2) × √3 |
Sustituye cateto_corto = hip/2 en cateto_largo = corto × √3. |
| Razón de lados 45-45-90 | 1 : 1 : √2 |
Ambos catetos son iguales (triángulo rectángulo isósceles); la hipotenusa es cateto × √2. |
| Hipotenusa 45-45-90 | hypotenuse = leg × √2 |
Para cualquier triángulo rectángulo con catetos iguales, la hipotenusa es √2 ≈ 1.414 veces cualquiera de los catetos. |
| Cateto 45-45-90 desde la hipotenusa | leg = hypotenuse / √2 = hypotenuse·√2/2 |
Inverso — cuando conoces la hipotenusa, divide por √2. |
| 45-45-90 a partir de un cuadrado | half a square diagonally = 45-45-90 |
Un triángulo 45-45-90 es exactamente la mitad de un cuadrado cortado a lo largo de su diagonal. Por eso s² + s² = (s√2)² = 2s². |
| 30-60-90 a partir de un equilátero | half an equilateral cut by altitude |
Un triángulo 30-60-90 es la mitad de un triángulo equilátero cortado por su altura. De ahí provienen las razones 1 : √3 : 2. |
| sen(30°), cos(60°) | = 1/2 |
Se lee directamente del triángulo 30-60-90: opuesto/hipotenusa = corto/2 = (1)/(2) cuando corto = 1. |
| sen(60°), cos(30°) | = √3/2 |
Opuesto/hipotenusa = largo/2 = √3/2 en el triángulo 1:√3:2. |
| sen(45°), cos(45°) | = √2/2 ≈ 0.707 |
En un 45-45-90, el opuesto y el adyacente del ángulo son ambos catetos (ambos = 1), y la hipotenusa = √2. Por lo tanto, sen = 1/√2 = √2/2. |
| tan(30°) | = 1/√3 = √3/3 |
opuesto/adyacente = corto/largo = 1/√3. |
| tan(45°) | = 1 |
opuesto/adyacente = cateto/cateto = 1. |
| tan(60°) | = √3 |
opuesto/adyacente = largo/corto = √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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