Triangle calculator

Two ways to solve a triangle

Any triangle is fully determined once you know enough of its sides and angles, and there are several classic combinations that pin it down completely. This calculator covers the two most common starting points: knowing all three side lengths (SSS, "side-side-side"), or knowing two sides plus the angle trapped between them (SAS, "side-angle-side"). Both cases always describe exactly one possible triangle, which is what makes them reliable to calculate from.

Solving from three sides (SSS)

When all three sides are known, the area comes from Heron's formula, and each angle is found using the law of cosines rearranged to solve for the angle opposite a given side. Before calculating anything, the triangle inequality is checked — the sum of any two sides must be greater than the third side, or no such triangle can physically exist.

• Heron's formula: Area = √(s(s−a)(s−b)(s−c)), where s is half the perimeter
• Law of cosines: cos(A) = (b² + c² − a²) / (2bc), and similarly for the other two angles

Solving from two sides and the included angle (SAS)

When you know two sides and the angle between them, the law of cosines finds the third side directly, and the area comes from a simpler formula using the two known sides and the sine of the angle between them. The remaining two angles are then found using the law of cosines again, which avoids the ambiguity that can come from using the law of sines on its own.

• Third side: c² = a² + b² − 2ab·cos(C)
• Area: Area = ½ · a · b · sin(C)

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