🔺 Triangle Solver
Solve any triangle: SSS, SAS, ASA, AAS. Laws of Sines and Cosines with steps.
Solved triangle
How to Use the Triangle Solver
Choose which three values you know, enter them (angles in degrees), and solve. Every missing side and angle is computed with the rule named at each step.
Pick the case matching your known values: SSS, SAS, ASA, or AAS.
Enter the three values. For SAS, the angle must be between the two sides.
Read all six values plus area, with verification checks.
The Two Laws
Law of Sines: a/sin A = b/sin B = c/sin C = 2R. Each side is proportional to the sine of its opposite angle. Use it when you know an angle-side opposite pair plus one more value (ASA, AAS). Law of Cosines: c² = a² + b² − 2ab·cos C, the Pythagorean theorem generalized with a correction term for the angle. Use it for SSS (solve for angles) and SAS (solve for the third side). Rearranged: cos C = (a² + b² − c²)/(2ab).
Solution Paths by Case
SSS: Law of Cosines three times for the angles; verify they sum to 180°. SAS: Law of Cosines for the unknown side, then Law of Sines for one angle, then 180° minus the two known for the last. ASA/AAS: third angle = 180° − A − B, then Law of Sines twice for the unknown sides. Area: Heron's formula for SSS (s = perimeter/2, area = √(s(s−a)(s−b)(s−c))), or (1/2)ab·sin C when an angle is known.
Validity Checks
Not every input makes a triangle. SSS must satisfy the triangle inequality: each side shorter than the sum of the other two. Angles in ASA/AAS must sum to less than 180°. The solver validates inputs and reports the specific problem instead of a silent wrong answer. When inputs pass, the three cross-checks (angle sum, equal sine ratios, largest-side-largest-angle) confirm the solution.
Frequently Asked Questions
What combinations of values can solve a triangle?
SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), and AAS (two angles and a non-included side). SSA (two sides and a non-included angle) is ambiguous and can yield 0, 1, or 2 triangles. You need at least one side: AAA gives shape but not size.
When do I use the Law of Sines vs the Law of Cosines?
Law of Sines (a/sin A = b/sin B = c/sin C) for ASA, AAS, and SSA cases: it connects sides to their opposite angles. Law of Cosines (c² = a² + b² − 2ab·cos C) for SSS and SAS: it finds a third side from two sides and the included angle, or an angle from three sides.
What is the ambiguous SSA case?
Given two sides and a non-included angle, the given side may swing to two positions, giving two valid triangles, or it may be too short to reach, giving none. This solver handles SSS, SAS, ASA, and AAS, which are unambiguous, and warns you away from SSA input.
How do I check a triangle solution?
Three checks: angles sum to exactly 180°, the Law of Sines ratios a/sin A, b/sin B, c/sin C are all equal, and the largest side is opposite the largest angle. This solver displays all three verifications.
What is the difference from the right triangle calculator?
The right triangle calculator uses the Pythagorean theorem and only works when one angle is exactly 90°. This solver handles any triangle using the Laws of Sines and Cosines. If your triangle is right-angled, both give the same answer.