Law of Sines Calculator
Enter two angles + one side (ASA/AAS), or two sides + a non-included angle (SSA). The calculator applies a / sin A = b / sin B = c / sin C, shows every step, and draws the triangle. The SSA mode handles the ambiguous case honestly — it reports no solution, one solution, or two solutions and lists both when they exist.
Worked example — the classic ambiguous case
A = 30°, a = 8, b = 15
First compute the altitude h = b·sin A = 15·sin 30° = 15 × 0.5 = 7.5.
Since h < a < b (7.5 < 8 < 15), there are two valid triangles:
| Solution | B | C | c |
|---|
| 1 (acute B) | 69.6359° | 80.3641° | 15.7743 |
| 2 (obtuse B) | 110.3641° | 39.6359° | 10.2065 |
Angle B comes from sin B = b·sin A / a = 15·0.5/8 = 0.9375, so B₁ = arcsin(0.9375) ≈ 69.6359° and B₂ = 180° − B₁ ≈ 110.3641°. Then C = 180° − A − B and c = a·sin C / sin A.
Decision rule — how SSA decides 0, 1, or 2
Let h = b·sin A (the altitude from C to the base).
- A ≥ 90°: one solution if a > b, otherwise none.
- A < 90°: none if a < h · one right triangle if a = h · two if h < a < b · one if a ≥ b.
Try a = 5 with the same A = 30°, b = 15: a = 5 < h = 7.5 → no solution (the side never reaches the base).
What this tool does not do: it never classifies a fully-specified triangle (three sides → right/acute/obtuse) — that check is
Is It a Right Triangle?. It never uses the
law of cosines (SSS/SAS) — it uses
only the law of sines.
Frequently asked questions
What is the law of sines?
The law of sines states that in any triangle, a / sin A = b / sin B = c / sin C, where lowercase letters are sides and uppercase letters are the angles opposite them. This common ratio equals 2R, twice the triangle's circumradius. It lets you find unknown sides from known angles and a known side, and vice versa.
When can I use the law of sines?
Use it for two cases: ASA/AAS (two angles and any one side known) and SSA (two sides and a non-included angle known). ASA/AAS always gives exactly one triangle. SSA is the ambiguous case and may give zero, one, or two valid triangles.
What is the ambiguous case of the law of sines?
The ambiguous case happens in SSA: given angle A, side a opposite it, and side b, two different triangles can satisfy the same inputs. For example, A = 30°, a = 8, b = 15 yields two solutions: (B ≈ 69.64°, C ≈ 80.36°, c ≈ 15.7743) and (B ≈ 110.36°, C ≈ 39.64°, c ≈ 10.2065). An honest calculator must report both, or honestly report zero when none exist.
How do you know whether SSA has no, one, or two solutions?
Compute the altitude h = b · sin A, then compare the side a. If A ≥ 90°: one solution when a > b, otherwise none. If A < 90°: no solution when a < h; exactly one right triangle when a = h; two solutions when h < a < b; one solution when a ≥ b. For instance, A = 30°, a = 5, b = 15 gives h = 7.5 > a, so there is no solution — the side never reaches the base.
Law of sines or law of cosines — which one do I need?
Use the law of sines for ASA, AAS, and SSA (it connects one side directly to its opposite angle). Use the
law of cosines for SSS (three sides) and SAS (two sides with the included angle). This calculator uses only the law of sines; a triangle classifier for three given sides is the Is It a Right Triangle? checker.
How do I check the answer by hand?
Pick any solution and verify two things: the angles sum to 180°, and the ratios a/sin A, b/sin B, c/sin C are all equal. For the worked example above, solution 1 gives 8/sin 30° = 16, 15/sin 69.6359° ≈ 16, and 15.7743/sin 80.3641° ≈ 16 — all equal.
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