About the Triangle Calculator
This triangle calculator solves a triangle from two sides and the angle between them, returning the third side, the remaining angles, and the area. It applies the law of cosines and the law of sines, which together solve any triangle, not just right-angled ones.
The formula
c² = a² + b² − 2ab·cos C and Area = ½ab·sin Ca and b are the known sides, C the included angle between them, and c the side opposite that angle. The law of cosines generalises the Pythagorean theorem to any triangle.
How to use this calculator
- 1Enter your Side a. The field starts at
5, which you can overwrite. - 2Enter your Side b. The field starts at
7, which you can overwrite. - 3Enter your Included Angle C (°). The field starts at
60, which you can overwrite. - 4Read the result straight away — it recalculates as you type, so there is no button to press. Use Share to copy a link that reopens the page with your exact numbers filled in.
Worked example
| Input | Value |
|---|---|
| Side a | 5 |
| Side b | 7 |
| Included Angle C (°) | 60 |
Result
Side c: 6.2450
Area: 15.1554
Perimeter: 18.2450
Angle A: 43.90° B: 76.10°
Those are the values the page loads with, so you can reproduce this result yourself and then change one field at a time to see what drives the outcome.
Understanding your result
The law of cosines reduces to Pythagoras when C is 90 degrees, because cos 90° = 0 and the correction term vanishes. For acute angles cosine is positive, so c comes out shorter than the Pythagorean value; for obtuse angles it is negative and c comes out longer. That single term captures the entire effect of the angle.
Two sides and the included angle always determine a unique triangle, which is why this configuration is reliable. Two sides and a non-included angle can produce two valid triangles — the ambiguous case — and three angles determine shape but not size. Knowing which combinations pin a triangle down is half of solving them.
Things worth knowing
- Angles in any triangle sum to 180 degrees, which is the check to run on your answer.
- The largest angle always sits opposite the longest side.
- The triangle inequality must hold: any two sides must sum to more than the third.
- The area formula ½ab·sin C needs no height, which makes it far more convenient than ½ base × height when you know an angle.
- Heron's formula gives the area from three sides alone with no angle required.
Frequently asked questions
What is the law of cosines?+
A generalisation of the Pythagorean theorem to any triangle: c² = a² + b² − 2ab·cos C. When C is 90 degrees the cosine term disappears and it reduces to a² + b² = c².
How do I find the area without knowing the height?+
Use ½ab·sin C with two sides and the included angle. If you know all three sides instead, Heron's formula gives the area directly.
Can any three lengths form a triangle?+
No. The triangle inequality requires that any two sides sum to more than the third. Lengths of 1, 2, and 5 cannot close into a triangle.
What is the ambiguous case?+
When you know two sides and an angle not between them, two different triangles can satisfy the data. Two sides with the included angle, as used here, always give a single unique triangle.