30-60-90 und 45-45-90 — einmal merken, sofort losen
Geprüft von [email protected], Geometry Calculator Developer & Online Math Educator Zuletzt aktualisiert am May 21, 2026
Zwei rechtwinklige Dreiecke treten in Geometrie, Trigonometrie und Physik so häufig auf, dass es sich lohnt, die Seitenverhältnisse auswendig zu lernen: das 30-60-90-Dreieck (Seiten 1 : √3 : 2) und das 45-45-90-Dreieck (Seiten 1 : 1 : √2). Sobald du die Winkel erkennst, kannst du die Seiten ablesen, ohne zu rechnen – das spart jedes Mal eine Pythagoras-Berechnung.
| Name | Formel | Hinweise |
|---|---|---|
| 30-60-90 Seitenverhältnis | 1 : √3 : 2 |
Gegenüber den Winkeln 30°, 60°, 90° der Reihe nach. Kurze Kathete, lange Kathete, Hypotenuse. |
| 30-60-90 Lange Kathete | long_leg = short_leg × √3 |
Die Seite gegenüber 60° ist √3 mal die Seite gegenüber 30°. |
| 30-60-90 Hypotenuse | hypotenuse = 2 × short_leg |
Die Hypotenuse ist immer genau doppelt so lang wie die kürzeste Seite (gegenüber 30°). |
| 30-60-90 Kurze Kathete aus Hypotenuse | short_leg = hypotenuse / 2 |
Umkehrung: Wenn du die Hypotenuse kennst, ergibt die Hälfte davon die Seite gegenüber 30°. |
| 30-60-90 Lange Kathete aus Hypotenuse | long_leg = (hypotenuse / 2) × √3 |
Setze kurze_Kathete = hyp/2 in lange_Kathete = kurz × √3 ein. |
| 45-45-90 Seitenverhältnis | 1 : 1 : √2 |
Beide Katheten sind gleich (gleichschenklig-rechtwinkliges Dreieck); die Hypotenuse ist Kathete × √2. |
| 45-45-90 Hypotenuse | hypotenuse = leg × √2 |
Bei jedem rechtwinkligen Dreieck mit gleichen Katheten ist die Hypotenuse √2 ≈ 1,414 mal eine der Katheten. |
| 45-45-90 Kathete aus Hypotenuse | leg = hypotenuse / √2 = hypotenuse·√2/2 |
Umkehrung – wenn du die Hypotenuse kennst, teile durch √2. |
| 45-45-90 aus Quadrat | half a square diagonally = 45-45-90 |
Ein 45-45-90-Dreieck ist genau die Hälfte eines Quadrats, das entlang seiner Diagonalen geschnitten wurde. Daher gilt s² + s² = (s√2)² = 2s². |
| 30-60-90 aus Gleichseitigem | half an equilateral cut by altitude |
Ein 30-60-90-Dreieck ist die Hälfte eines gleichseitigen Dreiecks, das durch seine Höhe geteilt wird. Daher stammen die Verhältnisse 1 : √3 : 2. |
| sin(30°), cos(60°) | = 1/2 |
Direkt aus dem 30-60-90-Dreieck abgelesen: Gegenkathete/Hypotenuse = kurz/2 = (1)/(2) wenn kurz = 1. |
| sin(60°), cos(30°) | = √3/2 |
Gegenkathete/Hypotenuse = lang/2 = √3/2 im 1:√3:2-Dreieck. |
| sin(45°), cos(45°) | = √2/2 ≈ 0.707 |
In einem 45-45-90-Dreieck sind Gegenkathete und Ankathete des Winkels beide Katheten (beide = 1), und die Hypotenuse = √2. Also sin = 1/√2 = √2/2. |
| tan(30°) | = 1/√3 = √3/3 |
Gegenkathete/Ankathete = kurz/lang = 1/√3. |
| tan(45°) | = 1 |
Gegenkathete/Ankathete = Kathete/Kathete = 1. |
| tan(60°) | = √3 |
Gegenkathete/Ankathete = lang/kurz = √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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