Solve triangles using the law of sines. Handles AAS, ASA, and SSA configurations including the ambiguous case with 0, 1, or 2 solutions.
The **Law of Sines Calculator** solves triangles using the relation a/sin(A) = b/sin(B) = c/sin(C). It accepts three standard configurations: AAS (two angles and a non-included side), ASA (two angles and the included side), and the notoriously tricky SSA case (two sides and a non-included angle) where zero, one, or two valid triangles may exist.
The ambiguous SSA case is the most interesting feature of this calculator. When two sides and a non-included angle are given, the calculator determines whether the data produces zero solutions (the shorter side cannot reach the opposite side), exactly one solution, or two distinct solutions. When two solutions exist, both are fully computed and displayed side by side so you can compare them.
For every valid solution, the calculator outputs all three sides, all three angles, the area (by Heron's formula), the perimeter, the common sine-ratio constant (a/sin A), the circumradius (R = ratio/2), the inradius, and a triangle-type classification. Side-proportion bars provide a visual comparison of the three sides. A detailed comparison table shows all properties for one or both solutions when the ambiguous case arises.
A reference table explains the four conditions of the SSA case with clear thresholds and highlights which condition matches your current input. Eight preset buttons load common configurations — including classic 30-60-90 and 45-45-90 triangles, ASA examples, and carefully chosen SSA cases that demonstrate zero-solution, one-solution, and two-solution scenarios.
Law of Sines Calculator — Triangle Solver with Ambiguous Case helps you avoid repetitive setup mistakes when solving trigonometric and coordinate-geometry problems. Instead of recalculating conversions, signs, and edge cases by hand, you can test inputs immediately, inspect intermediate values, and confirm final answers before submitting work or using numbers in downstream calculations. It surfaces key outputs like Side a, Side b, Side c in one pass.
a/sin(A) = b/sin(B) = c/sin(C). In SSA: sin(B) = b·sin(A)/a. If sin(B) > 1, no solution; if sin(B) = 1, one right triangle; if sin(B) < 1, check B₁ and B₂ = 180°−B₁ for the ambiguous case.
Result: Two solutions: B₁ ≈ 45.58°, B₂ ≈ 134.42°
Using a=7, b=10, A=30°, the calculator returns Two solutions: B₁ ≈ 45.58°, B₂ ≈ 134.42°. This example mirrors the calculator's live computation flow and is useful for checking manual steps and unit handling.
This calculator is tailored to law of sines calculator — triangle solver with ambiguous case workflows, including common input modes, unit handling, and special-case behavior. It is designed for fast checking during homework, exam preparation, technical drafting, and coding tasks where trigonometric consistency matters.
Use the primary result together with supporting outputs to verify direction, magnitude, and validity. Cross-check against known identities or geometric constraints, and confirm that angle ranges, sign conventions, and domain restrictions are satisfied before using the numbers elsewhere.
A reliable way to improve is to solve once manually, then verify with the calculator and explain any mismatch. Repeat this on varied examples and edge cases. The built-in preset scenarios for quick trials, comparison tables for side-by-side validation, visual cues that make trends and quadrants easier to read help you build pattern recognition and reduce sign or conversion errors over time.
The law of sines states that a/sin(A) = b/sin(B) = c/sin(C), meaning the ratio of each side to the sine of its opposite angle is constant for any triangle. Use this as a practical reminder before finalizing the result.
The ambiguous case occurs in SSA (two sides + non-included angle). Depending on the side lengths and angle, there may be 0, 1, or 2 valid triangles.
When b·sin(A) < a < b (side a is longer than the altitude but shorter than side b), there are two valid positions for the triangle, yielding two solutions. Keep this note short and outcome-focused for reuse.
When a < b·sin(A), side a is too short to reach the opposite side, so no triangle can be formed. Apply this check where your workflow is most sensitive.
The constant a/sin(A) = b/sin(B) = c/sin(C) equals 2R, where R is the circumradius — the radius of the circle passing through all three vertices. Use this checkpoint when values look unexpected.
The law of sines uses angle-side ratios and works best for AAS/ASA/SSA. The law of cosines relates all three sides to one angle and works best for SAS and SSS configurations.