最終更新日: 2026年8月19日
不等辺三角形計算機
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Quick Answer
The scalene triangle calculator solves a triangle from three unequal side lengths. It uses the law of cosines to compute the three interior angles and Heron’s formula to compute the area, then reports the perimeter and semiperimeter. This makes it useful for SSS geometry problems where you need a complete picture of an irregular triangle rather than only one measurement.
A scalene triangle is solved from its three unequal sides by using the law of cosines for the angles and Heron’s formula for the area.
重要ポイント
- A scalene triangle has three unequal sides and three unequal angles.
- Three known sides are enough to solve the whole triangle.
- The law of cosines recovers the interior angles from side lengths.
- Heron’s formula gives the area from the same three sides.
- The triangle inequality must hold before any scalene solution is valid.
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数式
A = arccos((b² + c² - a²)/(2bc)), B = arccos((a² + c² - b²)/(2ac)), C = 180° - A - B, Area = √(s(s-a)(s-b)(s-c))
ここで:
- a=Side a(units)
- b=Side b(units)
- c=Side c(units)
- A=Angle opposite side a(degrees)
- B=Angle opposite side b(degrees)
- C=Angle opposite side c(degrees)
- s=Semiperimeter(units)
解法例
Solve the 5-6-7 scalene triangle
This is a classic all-different triangle that produces clean Heron and cosine-rule practice.
- 1Use the law of cosines to find A ≈ 44.415309° and B ≈ 57.12165°.
- 2Use angle sum to find C ≈ 78.463041°.
- 3Compute s = (5 + 6 + 7) / 2 = 9.
- 4Area = √(9·4·3·2) = √216 ≈ 14.696938.
Solve the 8-9-10 scalene triangle
All three sides are different, so each angle is different as well.
- 1Use the law of cosines to get A ≈ 49.458398° and B ≈ 58.751559°.
- 2Then C ≈ 71.790043°.
- 3Compute s = 13.5.
- 4Heron’s formula gives area ≈ 34.197039.
Solve the 13-14-15 scalene triangle
A larger triangle with a whole-number area that is still truly scalene.
- 1Use the law of cosines to get A ≈ 53.130102° and B ≈ 59.489763°.
- 2Then C ≈ 67.380135°.
- 3Compute s = 21.
- 4Heron’s formula gives area = 84 exactly.
はじめに
The Scalene Triangle Calculator solves a triangle when all three sides are known and all three sides are different. Because no two sides match, a scalene triangle also has three different interior angles. This calculator uses the law of cosines to recover the angle measures and Heron’s formula to compute the area from the same side lengths. That makes it useful when you need a complete triangle summary instead of only one quantity, whether you are working on geometry homework, layout sketches, or measurement checks with irregular triangles.
What makes a triangle scalene
A scalene triangle has three unequal sides and therefore three unequal angles. That is the defining feature. If two sides match, the triangle is isosceles instead. If all three sides match, it is equilateral.
The triangle inequality still comes first
Before solving any three-side triangle, check that the sum of any two sides is greater than the third side. If that fails, the sides cannot close to form a real triangle and no angle or area calculation should proceed.
a + b must be greater than c.
a + c must be greater than b.
b + c must be greater than a.
All three side lengths must also be positive.
Finding the angles with the law of cosines
With all three sides known, the law of cosines is the most direct way to recover the interior angles. It converts the side-length relationships into cosines of the corresponding angles, and then inverse cosine returns the angle measures.
Finding the area with Heron’s formula
Heron’s formula uses the semiperimeter s = (a + b + c) / 2 and then computes Area = √(s(s-a)(s-b)(s-c)). This is perfect for scalene triangles because it avoids needing a separate height measurement.
Worked example with sides 5, 6, and 7
The 5-6-7 triangle gives A ≈ 44.415309°, B ≈ 57.12165°, and C ≈ 78.463041°. Its semiperimeter is 9 and its area is √216 ≈ 14.696938. Because all three sides differ, the triangle is genuinely scalene.
Common scalene-triangle mistakes
One frequent mistake is assuming that any triangle with three side inputs counts as scalene. It does not; the sides must be different. Another is forgetting that angle labels must stay opposite their matching side labels when applying the law of cosines.
Do not skip the triangle-inequality check.
Do not call a triangle scalene if two sides are equal.
Do not swap side labels when computing the angles.
Do not round an angle too early before computing the last angle.
When to use a scalene solver
Use this calculator when your main goal is to understand the full triangle, not just one output like area or perimeter. It is especially helpful when you want all three angles in addition to area and side-based summary values.
Real-world applications
Scalene triangles appear in irregular roof sections, truss members, site layouts, navigation triangles, and distance problems drawn from three measured segments. In these settings, the triangle is rarely symmetric, so a full solver is more useful than a special-case formula.
Manual solving versus calculator use
By hand, it is wise to compute one or two examples to see how the cosine formulas map sides to angles. After that, a calculator helps with repeated work, decimal inputs, and consistency checks across several candidate triangles.
Start with the largest side; it should face the largest angle.
Use Heron’s formula after the angle work to cross-check the triangle.
If two sides are nearly equal, double-check whether you really wanted a scalene calculator.
Keep the full-angle precision until the final displayed result.
クイックリファレンスカード
Scalene triangle quick reference
クイックリファレンス • 不等辺三角形計算機
Use the law of cosines for the angles and Heron’s formula for the area.有効範囲: Three positive, unequal side lengths that satisfy the triangle inequality.
一般的な値
⚠ 注意
- •All three sides must be different for a scalene triangle.
- •The triangle inequality must hold.
- •Angles should add to 180 degrees after solving.
- •The largest side should match the largest angle.
プロのヒント
- →Compute the angle opposite the largest side first as a reasonableness check.
- →Use area and semiperimeter together to verify the triangle is not degenerate.
- →If two measured sides are nearly equal, inspect the data before assuming it is scalene.
- →Keep at least a few decimal places during manual cosine-rule work.
よくある質問
What is a scalene triangle?
A scalene triangle is a triangle with three different side lengths and three different angle measures.
Can this calculator be used if two sides are equal?
No. If two sides are equal, the triangle is isosceles rather than scalene, so this calculator returns an error.
Why does the calculator need all three sides?
This solver is based on an SSS setup. Three known sides are enough to recover the angles with the law of cosines and the area with Heron’s formula.
How is the area found?
The area comes from Heron’s formula after computing the semiperimeter s = (a + b + c) / 2.
How can I check whether the answer is reasonable?
The largest side should face the largest angle, the angles should add to 180 degrees, and the area should be positive.
Why is the triangle inequality important?
Without it, the three segments cannot close into a triangle, so there is no real area or interior-angle solution.
When should I use the scalene-triangle-area calculator instead?
Use the area-specific calculator when you mainly want Heron’s-formula outputs such as area, semiperimeter, inradius, or a selected altitude.