What is the SSA 'ambiguous case'?

When you know two sides and an angle opposite one of them, the numbers can be consistent with zero, one, or two different valid triangles.

Why did I get 'no triangle exists' for my SSS inputs?

The two shorter sides didn't add up to more than the longest one — the triangle inequality requires that any two sides together must exceed the third.

How is area calculated?

Via Heron's formula, which only needs the three side lengths.

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Math & Everyday

Triangle Solver

Solve any triangle from 3 known sides or angles, with area, perimeter, and the ambiguous SSA case handled correctly.

🔒 Runs entirely in your browser — nothing here is ever uploaded

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About the Triangle Solver

Solves a triangle given any 3 of its 6 elements (3 sides, 3 angles) using the Law of Sines and Law of Cosines, reporting all sides, angles, area, and perimeter — and correctly flags the ambiguous SSA case as 0, 1, or 2 valid triangles rather than guessing.

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How to use it

Enter any 3 of the 6 fields, including at least one side.
Leave the other 3 fields blank.
Read the solved sides, angles, area, and perimeter.

Formula

Law of Cosines: c² = a² + b² − 2ab·cos(C) — used for SSS and SAS. Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) — used for ASA/AAS and to test both candidate solutions in the ambiguous SSA case. Heron's formula: Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2.

Worked example

Sides a=3, b=4, c=5 (SSS): ∠A=36.87°, ∠B=53.13°, ∠C=90.00°, area=6, perimeter=12. For SSA, sides a=20, b=25 with ∠A=40° yields two distinct valid triangles.

Interpreting your result

Three angles alone (AAA) fix a triangle's shape but not its size, so at least one side is required. SSS is rejected if the two shorter sides don't add up to more than the longest one. SSA can produce zero, one, or two valid triangles, and this tool checks for and reports all outcomes explicitly.

Recommendations

• If your known angle sits between your two known sides, that's SAS — always exactly one answer.
• If your known angle is opposite one of your two known sides, that's the ambiguous SSA case — check whether one or two triangles are reported.
• Given two angles and any one side, it doesn't matter whether that side is between the angles (ASA) or not (AAS).

Frequently asked questions

Any 3 of the 6 elements, as long as at least one side is included: SSS, SAS, ASA, AAS, and SSA.