The law of sines calculator solves a triangle whenever you have an angle paired with the side opposite it. That covers AAS and ASA, where two angles and one side are known, and it covers SSA — two sides and an angle that is not between them. SSA is the case this page exists for, because it is the one where the data can legitimately describe two different triangles, and most calculators quietly return only one of them.
Arb Digital built this tool to make that ambiguity visible rather than hidden. Before the answer appears, the calculator compares your side against the triangle's altitude and reports which of four outcomes you are in: no triangle, one right triangle, two triangles, or one triangle. Note the boundary with our other geometry pages — the triangle area calculator returns area alone, and the law of cosines calculator handles SSS and SAS. This page solves for unknown sides and angles from a matched angle-side pair.
What This Law of Sines Calculator Does
In AAS / ASA mode, enter two angles and any one side, and the calculator returns the third angle and both remaining sides. There is never any ambiguity here: two angles fix the third by subtraction, and the shape is completely determined, so a single side sets the scale and one answer follows.
In SSA mode, enter angle A, the side a opposite it, and a second side b. The calculator computes sin B, checks how many valid angles B can produce, and then reports every complete triangle that satisfies your input — including the second one, with its own angle C and side c, whenever a second one exists. That second column is the differentiator. A page that returns one answer to an SSA problem is giving you an incomplete solution roughly a third of the time.
How to Use It
- Choose your case. SSA if you have two sides and an angle outside them; AAS / ASA if you have two angles and any side.
- Enter angle A and side a together. They must be a matched pair — the angle and the side directly opposite it. Without a matched pair the law of sines cannot start.
- Fill the mode-specific box. Side b for SSA, angle B for AAS / ASA. The unused box is ignored.
- Pick degrees or radians. Both input and output use the selected unit.
- Read the tip line first in SSA mode. It tells you how many triangles exist and why, before you use any of the numbers.
The Formula: How It's Calculated
The law of sines states that the ratio of each side to the sine of its opposite angle is the same throughout a triangle:
a ÷ sin A = b ÷ sin B = c ÷ sin C
To find an unknown side, cross-multiply from a known pair: b = a · sin B ÷ sin A. To find an unknown angle from two sides, rearrange the other way: sin B = b · sin A ÷ a. That second rearrangement is where the ambiguity is born, and the reason is worth stating precisely — inverse sine returns only angles between −90° and 90°, but an interior triangle angle can be anywhere up to 180°. So for any sine value there are two candidate angles: the one your calculator returns, and its supplement, 180° minus that value. Both have the same sine. Only the geometry can decide which is real. The Wolfram MathWorld entry on the law of sines gives the standard statement and its circumradius form.
The Ambiguous SSA Case, Decided Properly
Here is the full decision procedure the calculator runs, and it is genuinely the most useful thing on this page. Start with angle A, its opposite side a, and the adjacent side b. Compute the altitude from the third vertex down to the line containing side c: h = b · sin A. That altitude is the shortest possible length side a could have and still reach the base. Now compare.
If A is acute, there are four outcomes. When a < h, side a is too short to reach the base and no triangle exists. When a = h exactly, it reaches at precisely one point, perpendicular to the base, giving one right triangle. When h < a < b, side a is long enough to reach the base but still shorter than b, so it can swing to meet the base at two different points — two triangles, one acute at B and one obtuse at B. When a ≥ b, only one of those two intersection points lies on the correct side of the vertex, so there is one triangle.
If A is obtuse or right, the analysis is shorter. Angle A is already the largest angle in the triangle, so its opposite side must be the longest. When a > b there is exactly one triangle; when a ≤ b there is none at all, because the data contradicts itself.
Working the example loaded above: A = 35°, a = 8, b = 12. The altitude h = 12 × sin 35° = 12 × 0.573576 = 6.883. Since 6.883 < 8 < 12, we are squarely in the two-triangle band. sin B = 12 × 0.573576 ÷ 8 = 0.860364, giving B = 59.363° or B = 120.637°. The first gives C = 85.637° and c = 13.906; the second gives C = 24.363° and c = 5.756. Both are real triangles with the exact same A, a and b. A page that reports only the 59° answer has silently discarded a valid solution.
How to Tell Which of the Two Triangles Is Yours
Mathematics cannot choose for you — both triangles satisfy the given measurements. The context has to. In practice the extra information usually already exists in the problem and just needs noticing.
Sometimes it is a stated angle range: if the problem says angle B is acute, or the diagram shows a clearly acute vertex, the obtuse solution is out. Sometimes it is a physical constraint: a distance c of 5.756 metres and one of 13.906 metres describe very different situations, and only one may fit the site. Sometimes the third angle rules it out — if C is known to exceed 90°, only the branch producing an obtuse C survives. And if genuinely nothing distinguishes them, the honest answer to the problem is both, stated as two solutions. If you can restructure the measurement to give three sides or an included angle instead, the law of cosines calculator is the better route precisely because it cannot be ambiguous.
Why AAS and ASA Are Never Ambiguous
Two angles determine the third instantly, since interior angles sum to 180°. That fixes the triangle's shape completely — every triangle with those three angles is a scaled copy of every other. Supply any one side and the scale is pinned down too, so exactly one triangle results. There is no supplement problem because no inverse sine of an unknown angle is ever taken; the third angle comes from subtraction, and the sides come from multiplication.
The only failure mode in AAS / ASA is a data error: if the two angles you entered already sum to 180° or more, no third angle is left and the calculator will tell you so. The same idea explains why the inverse functions differ so much in behaviour — see the arcsin calculator for why inverse sine's restricted range is the direct cause of the SSA problem, and the arccos calculator for the contrast.
The Circumradius Reading, and Where This Gets Used
The three equal ratios in the law of sines are not just equal to each other — they equal 2R, where R is the radius of the circle passing through all three vertices. So a ÷ sin A = 2R gives you the circumradius from a single matched pair, which is how the law connects triangle solving to circle geometry. Our circle calculator handles the circle's own measurements once you have that radius.
The practical home of the law of sines is triangulation. Surveyors measure a baseline of known length and two angles from its ends to a distant point, which is the ASA case, and the law of sines returns the distance without anyone crossing the intervening ground. The same method fixes positions in navigation and astronomy. The SSA case turns up in problems where a length is known but its orientation is not — a ladder of fixed length leaning against a wall at an unknown angle is the classic example, and it is a genuinely two-answer situation. The NIST Digital Library of Mathematical Functions covers the trigonometric definitions underpinning all of it, and the trigonometric functions calculator will check any individual sine value by hand.
Arb Digital publishes hundreds of free tools across maths, geometry, finance and unit conversion. No sign-up, no limits, nothing stored.
Browse All Free Tools Contact Arb DigitalCommon Mistakes to Avoid
- Reporting one answer to an SSA problem — check the altitude test first. Two triangles is a common outcome, not an exotic one.
- Forgetting the supplement — if sin B = 0.86, B could be 59.4° or 120.6°. Your calculator only ever offers the first.
- Using an angle and a side that are not opposite each other — the law of sines needs a matched pair. Adjacent pairs call for the law of cosines.
- Solving for the largest angle with inverse sine — if it is obtuse, inverse sine returns its acute supplement and the error is easy to miss. Find the small angles this way and get the large one by subtraction.
- Entering two angles that already sum to 180° or more — no third angle is left, so no triangle exists.
Related Free Tools From Arb Digital
Use the law of cosines calculator for SSS and SAS, the Pythagorean theorem calculator for right triangles, and the triangle area calculator when area alone is what you need. Check individual values with the trigonometric functions calculator, invert a ratio with the arcsin calculator, convert units with the angle converter, or browse the whole free online tools hub.
Frequently Asked Questions
It is the SSA situation, where you know two sides and an angle that is not between them. Because inverse sine returns only one of two possible angles, the same three measurements can describe two different triangles, one triangle, or none at all.
With angle A acute, compute the altitude h = b × sin A. If side a is shorter than h there is no triangle; if a equals h there is one right triangle; if a lies between h and b there are two triangles; and if a is at least as long as b there is exactly one.
Inverse sine returns angles only up to 90°, so any sine value has a second candidate at 180° minus that angle. Inverse cosine covers the whole 0° to 180° range an interior angle can occupy, so it never leaves a second candidate open.
a ÷ sin A = b ÷ sin B = c ÷ sin C. Each ratio also equals 2R, twice the radius of the circle passing through all three vertices of the triangle.
Mathematics cannot decide — both fit the data. Use context: a stated angle range, a diagram, a physical constraint on the missing length, or a known property of the third angle. If nothing distinguishes them, both are correct answers.
No. Two angles fix the third by subtraction and the shape is completely determined, so any single known side sets the scale and exactly one triangle results.
Yes. Switch the angle unit selector and every angle you enter and every angle returned will use radians rather than degrees.
This tool is provided for education and reference. Verify any result used in surveying, navigation or construction against an independent calculation.
