Triangle Solver
Solve a triangle from three sides (SSS) or two sides and the included angle (SAS). Get every side, angle, the area and perimeter — with the full working.
About this calculator
Choose a mode. SSS takes three sides and checks the triangle inequality, then finds each angle with the law of cosines and the area with Heron's formula. SAS takes two sides and the angle between them, finds the third side with the law of cosines, the area as ½·a·b·sin C, and the remaining angles with the law of sines.
How a triangle is solved
A triangle is fully determined once you know enough of its sides and angles. From three sides (SSS) the law of cosines gives each angle: cos A = (b² + c² − a²) / 2bc, and so on, while Heron's formula gives the area from the semiperimeter s = (a + b + c) / 2 as √(s(s−a)(s−b)(s−c)). From two sides and the included angle (SAS) the third side comes from c² = a² + b² − 2ab·cos C, the area is ½·a·b·sin C, and the remaining angles follow from the law of sines, sin A / a = sin B / b = sin C / c. The three sides must satisfy the triangle inequality — each side shorter than the sum of the other two — or no triangle exists.