Law of Sines Calculator
Solve any oblique triangle for missing sides, angles, and area using the Law of Sines. Supports AAS, ASA, and the full SSA ambiguous case with dynamic 2D geometric triangle plots.
If you have three side lengths (SSS) or two sides and the included angle (SAS), use our companion Law of Cosines solver.
Geometry Plot & Oblique Triangle
a/sin(A) = b/sin(B) = c/sin(C)Step-by-Step Law of Sines Derivation Formula: a / sin(A) = b / sin(B) = c / sin(C)
How to Solve a Triangle Using the Law of Sines
To solve an oblique triangle using the Law of Sines, set up the proportion a/sin(A) = b/sin(B) = c/sin(C). For AAS or ASA cases, first find the third angle using C = 180° - A - B, then cross-multiply to find missing sides: b = a·sin(B)/sin(A) and c = a·sin(C)/sin(A). For SSA cases, test the altitude h = b·sin(A) to check if 0, 1, or 2 triangles exist.
The Law of Sines Formula & Mathematical Ratio
The Law of Sines is a fundamental theorem of trigonometry that relates the lengths of the sides of any triangle (acute, obtuse, or right) to the sines of its angles:
Where $R$ is the circumradius of the triangle's circumscribed circle.
Solving AAS & ASA Triangle Cases
Whenever two angles are known, the third angle is immediately found via the Euclidean triangle angle-sum identity:
C = 180° − A − B
Once all three angles are established, missing sides are calculated using direct sine ratios without any ambiguity.
The Ambiguous Case (SSA): 0, 1, or 2 Triangles
When given two sides and a non-included acute angle $A$ (SSA), the altitude is $h = b \sin(A)$. The geometric possibilities depend on the length of side $a$:
Side $a$ is too short to reach the base line. No triangle exists.
Side $a$ meets the base line perpendicularly ($\sin B = 1, B = 90^\circ$).
Side $a$ can swing inward (obtuse $B_2 = 180^\circ - B_1$) or outward (acute $B_1$).
Side $a$ can only swing outward; the inward swing fails to form a valid triangle.
Comparing Law of Sines vs Law of Cosines
| Given Information | Recommended Method | Primary Formula |
|---|---|---|
| AAS or ASA | Law of Sines | a / sin(A) = b / sin(B) |
| SSA (Ambiguous) | Law of Sines | sin(B) = (b · sin(A)) / a |
| SAS | Law of Cosines | c² = a² + b² - 2ab cos(C) |
| SSS | Law of Cosines | cos(A) = (b² + c² - a²) / 2bc |
Step-by-Step Worked Examples (AAS & SSA Cases)
Solve triangle with A = 30°, a = 7, b = 10.
1. Altitude: h = 10 × sin(30°) = 10 × 0.5 = 5.0.
2. Test: 5.0 < a (7) < b (10) → Two triangles exist!
3. sin(B) = (10 × 0.5) / 7 = 5/7 ≈ 0.7143.
4. Triangle 1: B₁ = sin⁻¹(0.7143) = 45.58°, C₁ = 180° − 30° − 45.58° = 104.42°, c₁ = 7 × [sin(104.42°)/sin(30°)] = 13.56.
5. Triangle 2: B₂ = 180° − 45.58° = 134.42°, C₂ = 180° − 30° − 134.42° = 15.58°, c₂ = 7 × [sin(15.58°)/sin(30°)] = 3.76.
Common Pitfalls in Sine Ratio Calculations
Standard handheld calculators only return the principal acute angle from $\arcsin(\theta)$. In SSA cases, you must always test whether $B_2 = 180^\circ - B_1$ forms a valid second triangle.
Ensure your calculator is in Degree mode when working with degree inputs like $35^\circ$ or $65^\circ$.
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