SSA Triangle Calculator (Side-Side-Angle) — Ambiguous Case
Enter side a (opposite angle A), side b, and angle A. The calculator uses the law of sines to find 0, 1, or 2 valid triangles — the classic ambiguous case of triangle solving.
°
Ambiguous case — two triangles exist for these SSA inputs
Solution 1 — B acute
Solution 2 — B obtuse
- 1
sin B via law of sines
b × sin(A) ÷ a = 9 × sin(45°) ÷ 7 = 0.909137 - 2
Angle B = arcsin(sin B)
arcsin(0.909137) = 65.3864° - 3
Angle C = 180° − A − B
180° − 45° − 65.3864° = 69.6136° - 4
Side c via law of sines
a × sin(C) ÷ sin(A) = 9.2794
How does this calculator work?
SSA gives 0, 1, or 2 triangles. Compute sin(B) = b sin(A) / a — if > 1, no solution; otherwise B₁ = arcsin and B₂ = 180° − B₁ each yield a triangle when A + B < 180°. Find c = a sin(C) / sin(A) and Area = ½ bc sin(A) for each valid case. This is the only triangle type with an ambiguous case.
Formula
How this is calculated
The SSA (Side-Side-Angle) configuration is the "ambiguous case" of triangle solving: given two sides and an angle that is not between them, the triangle may have zero, one, or two solutions. The law of sines provides sin(B) = b sin(A) / a. If this value exceeds 1, no real angle B exists and no triangle is possible. If sin(B) = 1 exactly, there is one right-triangle solution with B = 90°. Otherwise, B₁ = arcsin(sin(B)) is one candidate and B₂ = 180° − B₁ is a second candidate; each is valid only if A + Bᵢ < 180° (otherwise angle C would be negative, which is geometrically impossible).
For each valid B, the remaining angle is C = 180° − A − B and the third side is c = a sin(C) / sin(A). Area follows from Area = ½ b c sin(A), the standard formula for two sides and their included angle. The inradius r = Area / s where s = (a + b + c) / 2 is the semi-perimeter, and the circumradius R = a / (2 sin A) is the same for both solutions because a and A are fixed.
Practically: if a ≥ b, there is always exactly one solution (B must be acute so B₂ would make A + B₂ > 180°). If a < b and A is acute, two solutions may exist — this is the true ambiguous case. If A ≥ 90° and a ≤ b, no triangle exists.
Frequently asked questions
When the known angle A is acute and side a is shorter than side b, the height h = b sin(A) matters: if a < h, no triangle exists; if a = h, one right triangle; if h ≤ a < b, two different triangles are both valid. This is the ambiguous case — unique to SSA, unlike SAS or ASA.
Geometrically, if side a is shorter than side b but long enough, swinging it from the end of side b can touch the opposite baseline at two points — creating two different triangles that share the same SSA data. The two solutions have the same circumradius but different areas.
Use additional context: a measured angle B (narrow or wide), a known approximate shape, or physical constraints. If the geometry requires an acute angle B, use Solution 1; if B is clearly obtuse, use Solution 2. When no extra info is available, report both solutions.
Also known as
TG we-Calculate Editorial Team. (2026). SSA Triangle Calculator (Side-Side-Angle) — Ambiguous Case [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/ssa-triangle-calculator
TG we-Calculate Editorial Team. "SSA Triangle Calculator (Side-Side-Angle) — Ambiguous Case." TG we-Calculate. 2026. https://we-calculate.com/calculator/ssa-triangle-calculator.
TG we-Calculate Editorial Team, "SSA Triangle Calculator (Side-Side-Angle) — Ambiguous Case," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/ssa-triangle-calculator
@misc{wecalculate_ssa_triangle_calculator, title = {SSA Triangle Calculator (Side-Side-Angle) — Ambiguous Case}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/ssa-triangle-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?