SSA Triangle Calculator

Enter two sides and a non-included angle to solve an SSA (side-side-angle) triangle with the Law of Sines — including the ambiguous case, where two different triangles can satisfy the same inputs.

Quick Facts

Law of Sines
a/sin A = b/sin B = c/sin C
Relates each side to the sine of its opposite angle.
SSA = the ambiguous case
Given two sides and a non-included angle
Unlike SAS or ASA, SSA can yield 0, 1, or 2 valid triangles.
Two-solution test
b·sin A < a < b
When true, both an acute and an obtuse angle B satisfy the inputs.

Your Results

Calculated
Number of Triangles
-
0, 1, or 2 valid solutions
Solution 1 (B, C, side c)
-
Angle B, angle C, and side c
Solution 2 (B, C, side c)
-
Second triangle, if the case is ambiguous
Height h = b·sin A
-
Threshold that determines the number of solutions

Ready

Enter sides a and b plus angle A, then press Calculate.

Formula and Method for the SSA Triangle (Law of Sines, Ambiguous Case)

SSA stands for Side-Side-Angle: you know the length of two sides and the measure of an angle that is not included between them (it is opposite one of the given sides). This calculator uses the Law of Sines, a/sin A = b/sin B = c/sin C, to find the remaining angles and side. Because the known angle is not between the two known sides, SSA is called the "ambiguous case" — depending on the numbers you enter, there may be no triangle, exactly one triangle, or two different triangles that both satisfy the given measurements.

How the calculation works

Given side a, side b, and angle A (opposite side a), the calculator first finds the height of the swing arc, h = b·sin A. This height tells you how many triangles are possible: if a < h, side a is too short to reach the opposite side and no triangle exists; if a = h, side a exactly reaches, forming a single right triangle; if a ≥ b, only one triangle is possible (the acute solution); and if h < a < b, two distinct triangles exist. When a solution exists, angle B is found from sin B = (b·sin A)/a. For each valid B, angle C = 180° − A − B, and the missing side is c = a·sin C / sin A. When two solutions exist, the second uses B₂ = 180° − B₁ (the obtuse supplement), which is only geometrically valid when the resulting angle sum stays under 180°.

Common mistakes

  • Confusing SSA with SAS: SSA's angle is opposite one of the given sides, not between them. If your angle sits between the two known sides, that is SAS and needs the Law of Cosines instead, not the Law of Sines.
  • Assuming only one triangle: unlike ASA or SAS, SSA data can describe two different triangles. Always check whether h < a < b before reporting a single answer.
  • Mixing degrees and radians: this calculator expects angle A in degrees; entering a radian value (like 0.70) instead of degrees (like 40) will produce a nonsensical result.

Real-world applications

  • Surveying and navigation use SSA (and the related Law of Sines cases) to find distances when only partial angle and distance data is available, such as triangulating a landmark from two bearings and one measured baseline.
  • Engineering and construction use it to solve for missing brace lengths or angles when two measurements and one non-included angle are known from a site survey.
  • Astronomy and navigation historically used SSA-style triangulation to compute distances to inaccessible points, such as ship-to-shore or star-angle calculations.
  • Recognizing the ambiguous case matters in any of these fields — reporting only one triangle when two are geometrically valid can lead to an incorrect distance or angle.

Frequently Asked Questions

What does SSA mean in triangle solving?
SSA stands for Side-Side-Angle: you know the lengths of two sides and the measure of an angle that is NOT between them (a non-included angle, opposite one of the given sides). This is different from SAS, where the known angle sits between the two known sides.
Why can an SSA triangle have two different solutions?
Because the angle is not between the two sides, the side opposite it can sometimes swing to two different positions and still connect the triangle, producing two valid triangles with different angles and a different third side. This is called the ambiguous case of the Law of Sines.
How do I know how many triangles are possible?
Compare side a (opposite the given angle A) to the height h = b·sin(A). If a < h, no triangle exists. If a = h or a ≥ b, exactly one triangle exists. If h < a < b, two triangles exist.
What formula does this calculator use?
It applies the Law of Sines, a/sin(A) = b/sin(B) = c/sin(C), to solve for angle B, then finds angle C = 180° − A − B and side c = a·sin(C)/sin(A) for each valid solution.

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