Triangle solver

Formula

How it works

With three sides known (SSS), the law of cosines finds the angles. With two sides and the angle between them known (SAS), the law of cosines finds the third side, then the law of sines finds the remaining angles. Heron's formula gives the area from the three sides.

Worked example

The same calculation with the default values:

• Known values Three sides (SSS)
• Side a 7
• Side b 9
• Side c, or the included angle (°) in SAS mode 12

1. cos C = (7² + 9² − 12²) ÷ (2 × 7 × 9) = -0.111111
2. A = asin(7 × sin 96.37937° ÷ 12) = 35.430945°
3. B = 180 − 96.37937 − 35.430945 = 48.189685
4. area = √(s(s−a)(s−b)(s−c)), s = 14 → area = 31.304952

Result: Side c: 12 · Angle A: 35.43094469 ° · Angle B: 48.1896851 °

Common mistakes

• Entering an impossible combination of sides or angles (e.g. two sides whose sum is less than the third, or angles that don't sum to 180°) - these don't describe a real triangle and the calculator cannot produce a meaningful result.
• Using the law of cosines when the simpler law of sines would do, or vice versa for the wrong given information (SSA vs SAS vs ASA) - each configuration has a natural matching formula, and forcing the wrong one leads to unnecessary complexity or an unsolvable step.
• Forgetting the ambiguous case (SSA) can have two valid triangles, one, or none - having two sides and a non-included angle doesn't always determine a unique triangle.