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SSA Triangle Calculator

The SSA (side-side-angle) configuration occurs when you know two sides and an angle opposite one of them—a setup that frequently leads to two possible triangles. This calculator applies the law of sines to resolve such cases, automatically detecting whether zero, one, or two valid triangles exist. Engineers, surveyors, and geometry students rely on this tool to navigate the ambiguous case without manual trial-and-error.

Last updated: May 13, 2026

Creators Jasmine J. Mah, BSc

Reviewers Mateusz Mucha and Anna Szczepanek

2,247 people find this calculator helpful

Law of Sines and SSA Triangle Resolution

When two sides and a non-included angle are given, the law of sines provides the foundation for finding unknowns:

a / sin(α) = b / sin(β) = c / sin(γ)

a, b, c — The three sides of the triangle
α, β, γ — Angles opposite to sides a, b, and c respectively

Understanding the SSA Ambiguity

The side-side-angle case differs fundamentally from SAS or ASA configurations because the known angle is not between the two known sides. This geometric arrangement can produce multiple solutions—or sometimes none at all.

No triangle exists if the given side opposite the known angle is too short relative to the other side and angle.
One unique triangle occurs when the geometry permits only a single valid configuration.
Two possible triangles arise when both an acute and an obtuse angle satisfy the law of sines, a scenario called the ambiguous case.

After calculating the unknown angle using sine law, you subtract it from 180° to find a potential second angle. The sum of this second angle and the originally given angle determines validity: if their total exceeds 180°, the second solution is rejected.

Practical Solving Steps

Identify that you have exactly two sides and an angle not between them. Use the law of sines to find the unknown angle opposite the known side:

Rearrange to solve: sin(β) = (b × sin(α)) / a
Calculate β using inverse sine (arcsin).
Check for a second angle: 180° − β.
Verify both angles by confirming their sum with the given angle stays below 180°.
Use the law of sines again to find the third side.

Always verify your final triangle by ensuring all angles sum to exactly 180° and all sides satisfy the law of sines consistently.

Common Pitfalls and Considerations

SSA triangle solving requires careful attention to angle validity and the ambiguous case.

Ambiguous case detection — When you calculate sin(β), remember that arcsin returns only one value (typically 0° to 90°). Always check whether 180° minus this angle also satisfies the original constraints. Both may be valid triangle solutions.
Right-angle and obtuse angles — If the given angle α is 90° or greater, no ambiguity exists—only one triangle can form. The known angle is already the largest, preventing a second valid configuration.
Impossible triangles — If sin(β) > 1 when applying the law of sines, the triangle cannot exist. This happens when the opposite side is too short relative to the other side and its opposite angle.
Angle sum verification — After solving, always confirm that all three angles sum to 180°. Rounding errors during calculation may introduce small discrepancies, but they should be negligible (within 0.01°).

Frequently Asked Questions

What is the difference between SSA and SAS triangles?

SAS (side-angle-side) provides the angle between the two known sides, guaranteeing a unique triangle. SSA gives two sides with an angle opposite one of them—no inclusion between them. This positional difference means SSA allows multiple solutions, while SAS never does. SAS is a valid congruence postulate; SSA is not.

When does the ambiguous case produce exactly two solutions?

Two solutions occur when the known angle is acute, the side opposite it is longer than the other known side but shorter than what the law of sines initially suggests for a single configuration, and the resulting second angle (180° minus the calculated angle) yields a valid geometry. If the sum of this second angle and the given angle is less than 180°, both triangles are legitimate.

How do I know if no SSA triangle can exist?

No triangle forms when applying the law of sines yields sin(β) > 1. Mathematically, this is impossible since sine values are bounded between −1 and 1. Geometrically, it means the opposite side is too short to reach from the endpoint of the known side at the given angle.

Can SSA triangles ever be congruent to other triangles?

Two SSA configurations may describe the same triangle, but SSA itself is not a congruence postulate in formal geometry. However, if two triangles are proven congruent by another method (SSS, SAS, ASA, AAS), they certainly match. The issue is that SSA alone cannot guarantee congruence without additional information.

Why is the law of sines the right tool for SSA?

The law of sines directly relates sides to their opposite angles: a/sin(α) = b/sin(β) = c/sin(γ). When you know two sides and a non-included angle, you can immediately solve for the angle opposite one of the known sides, then proceed to find the third angle and remaining side. Other laws (like the law of cosines) require different configurations.

What happens if I enter angles that don't sum to 180°?

An invalid entry indicates either an error in calculation or an impossible triangle configuration. Always re-verify your input values. If solving manually, recalculate the unknown angle using the law of sines and check for the ambiguous case before proceeding. The calculator will flag inconsistencies automatically.

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