30-60-90 と 45-45-90 — 一度覚えれば即座に解ける
[email protected], Geometry Calculator Developer & Online Math Educator が監修 最終更新 May 21, 2026
幾何学、三角法、物理学で頻繁に現れる2つの直角三角形の辺の比は覚えておく価値があります。30-60-90三角形（辺の比1：√3：2）と45-45-90三角形（辺の比1：1：√2）です。角度を認識すれば、計算せずに辺の長さを読み取ることができ、毎回ピタゴラスの計算を省けます。
| 名前 | 公式 | 備考 |
|---|---|---|
| 30-60-90の辺の比 | 1 : √3 : 2 |
それぞれ30°, 60°, 90°の角の対辺。短い脚、長い脚、斜辺。 |
| 30-60-90の長い脚 | long_leg = short_leg × √3 |
60°の対辺は30°の対辺の√3倍です。 |
| 30-60-90の斜辺 | hypotenuse = 2 × short_leg |
斜辺は常に最短辺（30°の対辺）のちょうど2倍です。 |
| 斜辺から30-60-90の短い脚を求める | short_leg = hypotenuse / 2 |
逆に、斜辺がわかれば、その半分が30°の対辺になります。 |
| 斜辺から30-60-90の長い脚を求める | long_leg = (hypotenuse / 2) × √3 |
短い脚 = 斜辺/2 を長い脚 = 短い脚 × √3 に代入します。 |
| 45-45-90の辺の比 | 1 : 1 : √2 |
両方の脚は等しい（直角二等辺三角形）。斜辺は脚 × √2です。 |
| 45-45-90の斜辺 | hypotenuse = leg × √2 |
等しい脚を持つ直角三角形では、斜辺はどちらかの脚の√2 ≈ 1.414倍です。 |
| 斜辺から45-45-90の脚を求める | leg = hypotenuse / √2 = hypotenuse·√2/2 |
逆に、斜辺がわかれば、√2で割ります。 |
| 正方形から45-45-90 | half a square diagonally = 45-45-90 |
45-45-90三角形は、正方形を対角線で切ったちょうど半分です。これが s² + s² = (s√2)² = 2s² となる理由です。 |
| 正三角形から30-60-90 | half an equilateral cut by altitude |
30-60-90三角形は、正三角形を高さで切った半分です。ここから比1：√3：2が生まれます。 |
| sin(30°), cos(60°) | = 1/2 |
30-60-90三角形から直接読み取れます：対辺/斜辺 = 短い脚/2 = (1)/(2)（短い脚=1の場合）。 |
| sin(60°), cos(30°) | = √3/2 |
1：√3：2の三角形では、対辺/斜辺 = 長い脚/2 = √3/2です。 |
| sin(45°), cos(45°) | = √2/2 ≈ 0.707 |
45-45-90では、角の対辺と隣辺はどちらも脚（どちらも=1）で、斜辺=√2です。したがって sin = 1/√2 = √2/2です。 |
| tan(30°) | = 1/√3 = √3/3 |
対辺/隣辺 = 短い脚/長い脚 = 1/√3。 |
| tan(45°) | = 1 |
対辺/隣辺 = 脚/脚 = 1。 |
| tan(60°) | = √3 |
対辺/隣辺 = 長い脚/短い脚 = √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.