Triangle Calculator (SSS, SAS, ASA, AAS, SSA)
Solve any triangle from any three elements. Law of Sines, Law of Cosines, and the ambiguous SSA case.
MATHSolve any triangle from any three known elements using Law of Sines and Law of Cosines. Supports SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the side between them), AAS (two angles and a non-included side), and the ambiguous SSA case (two sides and a non-included angle - can have 0, 1, or 2 valid solutions).
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Law of Cosines: c² = a² + b² - 2ab·cos(C). SSS and SAS resolve uniquely. ASA and AAS resolve once you find the third angle from A+B+C = 180°. SSA is the ambiguous case - it returns two solutions when sin(B) = b·sin(A)/a allows both an acute B and its supplement 180-B to satisfy the constraints. Output includes all six elements, area (½·a·b·sin(C)), perimeter, and triangle type (equilateral, isosceles, right, acute, obtuse, scalene).
Triangle Calculator (SSS, SAS, ASA, AAS, SSA)
Solve any triangle from any three known elements. Supports SSS, SAS, ASA, AAS, and SSA (ambiguous case) using the Law of Sines and Law of Cosines. Returns all six elements plus area, perimeter, and triangle type.
Sides
Angles
How Triangle Solving Works
Any triangle has six elements: three sides (a, b, c) and three angles (A, B, C). The angle opposite side a is A, opposite b is B, opposite c is C. To uniquely determine a triangle, you typically need three independent elements - and at least one must be a side (three angles alone only fix the shape, not the size).
Two main laws solve any triangle: Law of Sines a/sin(A) = b/sin(B) = c/sin(C), and Law of Cosines c² = a² + b² - 2ab·cos(C). SSS and SAS use Law of Cosines first to find a third element. ASA and AAS use Law of Sines after finding the third angle (A+B+C = 180°). SSA is the ambiguous case - it can have zero, one, or two valid triangles depending on the input.
The "ambiguous SSA case" occurs when the given angle is acute and the given side opposite that angle is shorter than the height of the would-be triangle. Two configurations can satisfy the constraints. Right and obtuse triangles avoid the ambiguity. When solving SSA by hand, always check both possible solutions and verify which (if any) makes geometric sense.
Results assume Euclidean (flat-plane) geometry. For spherical or hyperbolic triangles, different formulas apply.