Law of Sines Calculator

Background.

The Quanta Law of Sines calculator solves the SSA (side-side-angle) case of a triangle — you know one side, a second side, and the angle opposite the first side — using a/sin A = b/sin B = c/sin C. Unlike Quanta's general triangle solver, which deliberately restricts itself to the three unambiguous congruence cases (SSS, SAS, ASA), this calculator exists specifically to handle SSA, including its famous complication: SSA is the one classical input pattern that does not always determine a unique triangle. Depending on the numbers, zero, one, or two distinct triangles can satisfy the same three measurements, and this calculator returns the primary solution, tells you how many solutions actually exist, and lets you request the alternate solution when a second one is genuinely available.

SSA data shows up constantly in fields where you can measure a distance and a bearing angle but not a second angle directly. A ship's navigator sighting two landmarks knows the distance to one landmark, the angle to the other from the ship's heading, and the distance between the landmarks — an SSA triangle. A surveyor triangulating a property corner from a known baseline and a single measured angle faces the same setup. A fire-lookout tower operator reporting a smoke sighting by distance and bearing, cross-referenced against a second tower's known distance, is solving SSA. An artillery or aviation navigation problem involving a known range, a known baseline, and a single bearing angle is SSA as well. In every one of these cases, the ambiguous case is not a mathematical curiosity — it is a genuine, practical possibility that a careless calculator could silently paper over by picking one answer and hiding the fact that a second, equally valid answer exists.

The calculator handles this honestly. It computes h = b·sin(A) — the height of the triangle if angle A's opposite side swept down onto the baseline — and classifies your input using the classical rule: if the known side a is shorter than h, no triangle exists at all, and the calculator raises a clear error rather than returning a nonsensical result. If a equals h exactly, exactly one triangle exists, a right triangle with a 90° angle at B. If a is strictly between h and b, two distinct triangles satisfy the input, one with an acute angle B and one with an obtuse angle B — and the calculator's solution-preference option lets you inspect either one. If a is at least as large as b, exactly one triangle exists, and angle B is always acute.

If you already know you have SSS, SAS, or ASA data instead, Quanta's triangle solver and Law of Cosines calculator are the right tools — both guarantee a unique answer because those input patterns are never ambiguous. Reach for this calculator specifically when your known angle sits opposite one of your two known sides.

What is law of sines calculator?

The Law of Sines states that in any triangle, each side is proportional to the sine of its opposite angle, with the same constant of proportionality for all three: a/sin A = b/sin B = c/sin C = 2R, where R is the radius of the triangle's circumscribed circle. It is the natural tool whenever your known data pairs an angle with its opposite side plus one more measurement — here, a second side.

Given sides a and b and angle A (opposite a), the Law of Sines rearranges to sin B = (b·sin A) / a, which determines angle B up to the fundamental ambiguity of the sine function: sin(θ) = sin(180° − θ) for any angle θ, so both an acute value and its obtuse supplement are mathematically valid candidates for B. Whether both candidates correspond to an actual triangle depends on comparing a, b, and h = b·sin A. If a < h, neither candidate produces a valid triangle (angle sum would exceed 180°) — no solution. If a = h, the two candidates coincide at exactly 90° — one solution, a right triangle. If h < a < b, both candidates yield a valid third angle and a positive third side — two solutions, the ambiguous case. If a ≥ b, only the acute candidate yields a valid triangle — one solution.

Once angle B is resolved, angle C follows from the angle-sum identity (C = 180° − A − B) and side c follows from the Law of Sines again (c = a·sin C / sin A). The area follows from (1/2)·a·b·sin C, the same formula used throughout Quanta's other triangle calculators.

How to use this calculator.

1. 01Enter side a — the side directly opposite the angle you know.
2. 02Enter side b — the other known side.
3. 03Enter angle A in degrees — the known angle, opposite side a.
4. 04Leave the solution preference on "Acute" for the default, smaller-angle-B solution. If the calculator reports that two triangles fit (solutionCount = 2), switch to "Obtuse" to see the alternate, larger-angle-B triangle.
5. 05Read angle B, angle C, and side c together — they fully describe the resulting triangle alongside your original inputs.
6. 06Check solutionCount: 1 means your inputs determine a unique triangle (or you are looking at one of the two ambiguous solutions); 2 confirms you are in the genuinely ambiguous zone and both an acute and an obtuse triangle are valid.
7. 07If the calculator raises an error, no triangle exists for your inputs — side a is too short to reach the third vertex given the angle and side b you entered. Double-check your measurements, or confirm you have not mislabeled which angle is A.

The formula.

The Law of Sines follows from dropping a perpendicular height h from vertex C onto side c (or, equivalently, from any vertex onto the opposite side). In the right triangle formed on one side of that perpendicular, h = b·sin A; in the right triangle formed on the other side, h = a·sin B. Setting the two expressions for h equal gives a·sin B = b·sin A, which rearranges to a/sin A = b/sin B. Repeating the construction from a different vertex gives the third ratio, c/sin C, equal to the same value — and that shared value equals 2R, where R is the circumradius, a fact provable by inscribing the triangle in its circumscribed circle and applying the inscribed-angle theorem. The underlying idea — that corresponding sides of similar configurations are proportional — is the same principle Euclid proves for equiangular triangles in Book VI, Proposition 4 of the Elements.

The SSA ambiguous case arises because the equation sin B = (b·sin A)/a can have two valid solutions for B in the range (0°, 180°): an acute angle θ and its supplement 180° − θ, since sine is positive and symmetric about 90° across that whole range. Both are mathematically valid arcsine outputs; the question is whether each one closes into a real triangle once you add angle A and check that the sum stays under 180°. Write h = b·sin A for the height of the configuration. If a < h, side a is too short to reach the baseline at all — neither candidate for B produces a valid triangle, so zero solutions exist. If a = h, the two candidates coincide exactly at B = 90°, giving one right-triangle solution. If h < a < b, both the acute candidate and its obtuse supplement keep angle A + angle B under 180°, so both close into valid, geometrically distinct triangles — two solutions. If a ≥ b, the obtuse candidate always pushes A + B over 180° (because a ≥ b implies A ≥ B, and an obtuse B combined with A ≥ B would itself already exceed 90° + 90°), so only the acute candidate survives — one solution. This calculator computes all four branches explicitly rather than blindly taking the first arcsine result, and raises a descriptive error for the zero-solution case instead of ever returning a not-a-number result.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Weisstein, Eric W. "Law of Sines." MathWorld — A Wolfram Web Resource. States a/sin A = b/sin B = c/sin C = 2R and its derivation via the circumradius.
2. [2]Euclid, Elements, Book VI, Proposition 4 — in equiangular triangles, the sides about the equal angles are proportional; the classical proportionality principle underlying the Law of Sines (David E. Joyce edition, Clark University).
3. [3]NIST Digital Library of Mathematical Functions, §4.14 — definitions and periodicity of the circular (trigonometric) functions.
4. [4]Coxeter, H. S. M. Introduction to Geometry, 2nd edition. Wiley, 1969 — classic reference textbook covering the Law of Sines and the ambiguous SSA case.