Law of Sines Calculator

Solve any triangle using the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Covers AAS, ASA, and SSA (ambiguous case with 0, 1, or 2 solutions). Computes circumradius, area, and shows surveying/navigation applications.

Side b
—
Angle C (°) —
Side c —
Common ratio a/sin(A) —
Extended More scenarios, charts & detailed breakdown ▾
Side b
—
Angle C (°) —
Side c —
Area = ½ab sin(C) —
Professional Full parameters & maximum detail ▾

Sine Rule Results

Sine rule ratio a/sin(A) —
Circumradius R = a/(2 sin A) —
Area = ½ab sin(C) —

Applications

Navigation/surveying note —

How to Use This Calculator

  1. For AAS: enter two angles and the side opposite one of them.
  2. For SSA: enter two sides and the angle opposite one — check for ambiguous case.
  3. For ASA: enter two angles and their included side.
  4. Use Professional for circumradius, area, and application notes.

Formula

a / sin(A) = b / sin(B) = c / sin(C) = 2R

Area = ½ · a · b · sin(C) | Circumradius R = a / (2 sin A)

Example

AAS: A=40°, B=60°, a=7 → ratio=7/sin40°≈10.89 → b≈9.43, C=80°, c≈10.73

Frequently Asked Questions

  • The Law of Sines states that for any triangle, a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius. It relates each side to the sine of its opposite angle.
  • Use the Law of Sines when you know AAS, ASA, or SSA. Use the Law of Cosines for SAS or SSS cases. The Law of Sines requires at least one angle-side opposite pair.
  • When given two sides and a non-included angle (SSA), there may be 0, 1, or 2 valid triangles. If sin(B) = b·sin(A)/a > 1 there is no triangle; if sin(B) = 1 there is one (right) triangle; otherwise there may be two.
  • R = a / (2·sin(A)). The circumradius is the radius of the circle passing through all three vertices of the triangle.
  • It is used in surveying (triangulation to find distances), navigation (calculating position from bearings), and engineering (force vector analysis).

Related Calculators

📐 Law of Cosines △ Triangle Calculator 📐 Sin Cos Tan Calculator a²+b² Pythagorean Theorem
Sources & References (5) ▾
  1. Calculus — James Stewart, Appendix D: Trigonometry — Cengage Learning
  2. OpenStax Algebra and Trigonometry — 10.1 Non-right Triangles: Law of Sines — OpenStax
  3. Khan Academy — Law of Sines — Khan Academy
  4. NIST DLMF — Trigonometric Functions — NIST
  5. MIT OCW 18.01 — Trigonometric Functions and Applications — MIT OCW