Solve a triangle from two angles and a side, or from two sides and a non-included angle — including the classic ambiguous SSA case.
Choose a mode, enter your known angles and sides, and click Calculate
This law of sines calculator solves a triangle whenever you know two angles and one side (the AAS or ASA case), or two sides and a non-included angle (the trickier SSA case). The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C) for every triangle, linking each side to the sine of the angle opposite it. That single ratio is enough to unlock every missing angle and side — except in the SSA case, where the same numbers can sometimes describe two genuinely different triangles at once. This tool handles that ambiguity automatically, showing every valid solution it finds instead of silently picking one.
Given two angles and a side, it finds the third angle by subtracting from 180° and then applies the Law of Sines ratio to compute the two remaining sides. Given two sides and a non-included angle, it computes the sine of the second angle, checks whether that value is even possible, and then tests both the acute and obtuse candidate angles to see which one (or both) complete a valid triangle.
Trigonometry and precalculus students working through Law of Sines problem sets, surveyors and navigators triangulating a position from two known bearings, engineers checking truss or bracket geometry, and anyone who has partial triangle measurements from a real-world sketch or survey and needs the rest solved reliably — especially in the SSA case, which is notoriously easy to get wrong by hand.
Many real triangles can't be measured completely by hand — you might be able to sight two angles and a baseline distance, or measure two lengths and one angle, but not every side directly. The Law of Sines bridges that gap using only trigonometric ratios, and understanding when it applies (versus when the Law of Cosines is required) is a foundational skill in trigonometry, surveying, and navigation.
Surveyors use it to compute distances across terrain that can't be measured directly, from two angle sightings and one known baseline. Navigators use it for triangulation between landmarks or radio beacons. Architects and engineers use it to verify truss, roof, or bracket angles. Students and teachers use it to check homework on oblique (non-right) triangles quickly and see the ambiguous case worked out in full.
How two angles and a side, or two sides and an angle, solve the rest of the triangle
Because a/sin(A), b/sin(B), and c/sin(C) are all equal, knowing any one full pair plus one more value solves the whole triangle.
SSA input can match zero, one, or two real triangles, since sin(θ) and sin(180°−θ) are identical — this calculator checks both possibilities.
Every triangle's three interior angles add up to exactly 180°, which is how the third angle is always found first.
From choosing a mode to reading every valid solution
Select the AAS/ASA tab if you know two angles and a side, or the SSA tab if you know two sides and a non-included angle.
Type the known angle(s) in degrees and the known side length(s) into the fields for that mode.
The calculator applies the Law of Sines ratio to solve for every missing angle and side.
In SSA mode, the calculator tests both candidate angles and shows every solution — zero, one, or two — that actually forms a valid triangle.
Review the completed angles and sides for each valid solution shown in the results panel.
Solving a triangle with a = 8, b = 10, and Angle A = 35° — a case with two valid solutions
A surveyor measures two sides of a triangular plot, a = 8 and b = 10, and the angle opposite side a as A = 35°. How many triangles fit these measurements, and what are their remaining angles and sides?
Explanation: Both triangles genuinely satisfy a = 8, b = 10, A = 35° — one has a "short" third side (c₂ ≈ 2.61) and an obtuse angle B, the other has a "long" third side (c₁ ≈ 13.77) and an acute angle B. Without checking both candidates, it's easy to report only one and silently discard a mathematically valid triangle.
What each output means and how many solutions to expect
| Output | What It Represents | Typical Range / Notes |
|---|---|---|
| Angle C (AAS/ASA) | The third interior angle | 0° < C < 180°; invalid if A + B ≥ 180° |
| Side b, Side c (AAS/ASA) | The two unknown side lengths | Always positive; scales with side a |
| Number of SSA solutions | How many triangles fit the given a, b, A | 0 (no triangle), 1 (one solution), or 2 (ambiguous case) |
| Solution 1 (SSA) | The acute candidate for angle B and its resulting triangle | B₁ = asin(sin B) is always ≤ 90° |
| Solution 2 (SSA) | The obtuse supplement candidate, if it forms a valid triangle | B₂ = 180° − B₁ is always ≥ 90° |
Reading the results together: in AAS/ASA mode there is always exactly one solution (or none, if the angles are invalid). In SSA mode, treat the "number of solutions" line as the headline result — it tells you whether you're looking at a single unambiguous triangle or two equally valid ones.
Typical ranges: all angles must fall strictly between 0° and 180°, and the three angles of any single solution must sum to exactly 180°. All side lengths must be positive.
Manual verification: for any solution shown, check that a/sin(A) = b/sin(B) = c/sin(C) holds (all three ratios equal) and that A + B + C = 180° — both should match closely, aside from rounding in the last displayed decimal.
Where solving a triangle from angles and a side is genuinely useful
Check AAS, ASA, and ambiguous SSA problems step by step.
Compute an inaccessible distance from two angle sightings and one known baseline.
Locate a ship, aircraft, or landmark position from two bearing angles and a known distance.
Verify truss, gusset, or bracket angles when only partial triangle measurements are available.
Estimate a transmitter's location from two known angles relative to two receivers.
Find heights or distances across terrain that can't be measured directly.
Check oblique roof, truss, or facade angles against design specifications.
Compute distances to a landmark using two angle sightings from a moving vessel.
Solve angle-of-fire triangles where two angles and a baseline are known.
Model triangulated positioning problems that reduce to oblique triangles.
Resolve non-right vector triangles in force, velocity, or displacement problems.
Solve triangle geometry for procedural meshes, lighting angles, or hit-detection shapes.
See exactly when SSA measurements yield two valid triangles instead of one.
What this Law of Sines calculator does well, and where it doesn't apply
How to tell which formula applies to your known triangle values
| Known Values | Case Name | Formula to Use | Number of Solutions |
|---|---|---|---|
| Two angles + any side | AAS / ASA | Law of Sines | Always 1 (or 0 if angles invalid) |
| Two sides + non-included angle | SSA | Law of Sines (ambiguous case) | 0, 1, or 2 |
| Two sides + included angle | SAS | Law of Cosines | Always 1 |
| All three sides | SSS | Law of Cosines | Always 1 |
Summary: This Law of Sines calculator solves AAS, ASA, and the ambiguous SSA case instantly, showing every valid triangle it finds. Pair it with the Law of Cosines Calculator for SAS/SSS triangles, or the Triangle Solver for a general-purpose tool covering every case.
Common questions about the Law of Sines and the ambiguous case
Trusted educational references to go deeper on solving triangles
Explore other trigonometry and geometry tools