SSA Triangle Calculator (the ambiguous case)
Two sides and an angle that is not between them — the one case that can describe two different triangles, one, or none. This page tells you which, and gives both answers when there are two.
With sides a and b and angle A opposite a, the law of sines gives sin B = b·sin A ÷ a.
How the ssa triangle calculator (the ambiguous case) works
With sides a and b and angle A opposite a, the law of sines gives sin B = b·sin A ÷ a. That equation has two solutions between 0° and 180° — B and 180° − B — and both may produce a valid triangle, because the sine of an angle and of its supplement are equal. If b·sin A is larger than a, side a is simply too short to reach the base line and no triangle exists. If it is exactly equal, the triangle is right-angled at B and there is exactly one.
Formula: a² = b² + c² − 2bc·cos A; a / sin A = b / sin B = c / sin C; area = ½ab·sin C
Worked examples
| Inputs | Area | Note |
|---|---|---|
| a = 6, b = 8, A = 40° — two triangles fit | 23.7054 | the ambiguous case in action |
| a = 2, b = 10, A = 80° — none fit | — | side a cannot reach |
| a = 10, b = 6, A = 50° — one fits | 29.2733 | a longer than b, so only one answer |
FAQFrequently asked questions
Why does SSA sometimes give two answers?
Because sin B and sin(180° − B) are equal, so two different angles satisfy the law of sines. Both can form a real triangle when the known side is shorter than the other.
When is there only one triangle?
When the side opposite the known angle is the longer of the two, or when the second solution would push the angle total past 180°.
When is there no triangle at all?
When b·sin A is greater than a — the short side cannot reach the base line however it is swung.
Is SSA a congruence rule?
No, and that is exactly why: it does not guarantee a unique triangle, so it is not among SSS, SAS, ASA and AAS.
How do I know which of the two is mine?
The mathematics cannot tell you — you need one more measurement, or knowledge of whether the triangle is acute or obtuse.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.