SSA Triangle Calculator

The SSA Triangle Calculator resolves triangles where two sides and a non-included angle are known. Here's how to use it:

Step 1: Enter Side a — the side opposite to the known angle α.

Step 2: Enter Side b — the second known side.

Step 3: Enter Angle α — the angle opposite to side a (must be between 0° and 180°).

Step 4: Click "Solve Triangle". The calculator instantly tells you whether there is no valid triangle, exactly one solution, or two valid triangles (the ambiguous case).

Step 5: Review the complete angle and side breakdown for each valid solution.

The SSA Triangle Calculator is built on the Law of Sines:

a / sin(α) = b / sin(β) = c / sin(γ)

Where a, b, c are the side lengths and α, β, γ are their opposite angles.

To find angle β: sin(β) = b × sin(α) / a

— If sin(β) > 1: no real triangle exists. — If sin(β) = 1: exactly one right triangle. — If sin(β) < 1: two candidates exist for β. The supplementary angle (180° − β) may also yield a valid triangle if the remaining angle γ stays positive, producing the ambiguous (two-solution) case.

Once β is known, γ = 180° − α − β, and side c is resolved as: c = a × sin(γ) / sin(α)

The SSA (Side-Side-Angle) Triangle Calculator is an essential mathematical tool designed to help students, educators, and professionals instantly solve complex triangular geometries using the Law of Sines. While most triangle configurations (like Side-Side-Side or Side-Angle-Side) guarantee a single, unique solution, the SSA configuration is uniquely challenging because it introduces the 'ambiguous case.'

When you are given two sides and a non-included angle, the geometric constraints do not necessarily define one specific triangle. Depending on the exact measurements, the inputs can result in three different outcomes: no valid triangle can be formed (the side is too short to close the shape), exactly one unique right or oblique triangle exists, or two entirely different but equally valid triangles can be constructed. Determining which scenario applies requires manually calculating sine ratios and cross-checking angle limits.

Our straightforward calculator handles the heavy lifting by automatically detecting the ambiguous case. It instantly evaluates your inputs to determine whether zero, one, or two solutions are possible, and then calculates the complete set of missing angles and sides for every valid configuration. This makes it an invaluable resource for verifying homework, solving surveying problems, or exploring advanced trigonometry without the tedious manual calculations.