SSA triangle calculator (ambiguous case)
SSA is the ambiguous case of the law of sines: the same data can give two different triangles. This shows every possible solution.
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How to use it
- Enter known angle a, side a, opposite angle a and side b.
- The result appears straight away and updates as you edit.
- Copy the result, or copy a link that keeps the same values to share it.
How it is calculated
sin B = b × sin A / a. Above 1 there is no triangle. Otherwise B₁ = arcsin of it and B₂ = 180° - B₁, and each one with A + B < 180° is kept. Then C = 180° - A - B and c = a × sin C / sin A.
Example
Inputs: Known angle A = 40 °, Side a, opposite angle A = 8, Side b = 10. Result: Number of possible triangles: 2; Solution 1: angle B: 53.4641 °; Solution 1: angle C: 86.5359 °.
Limits and caveats
- Angle A must be opposite side a, and b is the other known side.
- Values very close to the borderline case can flip with small measuring errors.
Questions about this tool
Why is SSA called the ambiguous case?
The sine gives two candidate angles, B and 180 - B, and both can fit, producing two different triangles.
When is there only one solution?
When a is at least as long as b, or when sin B is exactly 1 and the triangle is right angled.
When is there no triangle?
When a is shorter than the height b × sin A, or when A is obtuse and a is not longer than b.