Law of Sines Calculator

The law of sines (also called the sine rule) says every side of a triangle divided by the sine of its opposite angle gives the same number. Know an angle and the side across from it, plus one more measurement, and the whole triangle follows. With A = 35°, B = 65° and a = 12, the ratio is 12 / sin 35° = 20.9214, which gives b = 18.9612 and c = 20.6035.

Three sister tools cover the neighboring jobs. The Triangle Calculator solves a triangle from any valid combination, including SSS and SAS where the sine rule cannot start. The Trigonometry Calculator evaluates sin, cos and tan on their own and handles right triangles with SOH-CAH-TOA. The Pythagorean Theorem Calculator is the fastest route to a missing side when one angle is exactly 90°. Use this page when you have a side paired with its opposite angle in an oblique (non-right) triangle, and especially when you need the ambiguous SSA case resolved properly.

How the law of sines works

In any plane triangle, label the vertices A, B and C, and label each side with the lowercase letter of the angle it faces: side a is opposite angle A, side b is opposite angle B, side c is opposite angle C. The rule is then:

Every one of those three fractions collapses to the same number, and that number equals 2R, the diameter of the circle drawn through all three vertices. The reason is simple: the perpendicular height dropped from vertex C can be written two ways, as b x sin A and as a x sin B. Setting them equal gives a / sin A = b / sin B, and repeating the argument from another vertex adds the third fraction.

Rearranged for the two things you normally want:

Solving for a side is safe. Solving for an angle is where the ambiguity creeps in, because the inverse sine hands back only the acute answer even when an obtuse one fits just as well.

Which case do you have?

The sine rule needs one complete pair, an angle together with the side opposite it, before it can move. That gives three usable input patterns:

• AAS (angle, angle, side): two angles and a side that is not between them. The third angle comes from the 180° sum, and the pair is complete. Exactly one triangle.
• ASA (angle, side, angle): two angles and the side between them. Again the third angle comes from the 180° sum, and that third angle is opposite the known side. Exactly one triangle.
• SSA (side, side, angle): two sides and an angle opposite one of them. Zero, one or two triangles, which is why it is called the ambiguous case.

SSS (three sides) and SAS (two sides with the angle between them) give no complete pair, so the sine rule cannot start. Use the law of cosines for the first step there, then switch back to the sine rule for the rest.

Worked example 1: AAS

A roof truss has a bottom angle A = 35°, a second angle B = 65°, and the member opposite A measures a = 12 ft. Work through it in four moves:

1. Third angle: C = 180° − 35° − 65° = 80°.
2. Common ratio: a / sin A = 12 / sin 35° = 12 / 0.573576 = 20.9214.
3. Side b: 20.9214 × sin 65° = 20.9214 × 0.906308 = 18.9612 ft.
4. Side c: 20.9214 × sin 80° = 20.9214 × 0.984808 = 20.6035 ft.

The perimeter is 12 + 18.9612 + 20.6035 = 51.5647 ft, and the area is 0.5 × 12 × 18.9612 × sin 80° = 112.0388 square feet. Heron's formula on the same three sides also returns 112.0388, so the solution checks out. The circumradius is 20.9214 ÷ 2 = 10.4607 ft.

The ratio really is constant

The table below takes the solved triangle above and divides each side by the sine of its own opposite angle. All three columns land on the same number, which is the whole point of the rule and a fast way to catch an arithmetic slip:

Worked example 2: ASA

Now the known side sits between the two known angles. Take A = 42°, B = 75°, and the side between them c = 20 in. The third angle is C = 180° − 42° − 75° = 63°, and that angle is opposite the side you already have, so the pair is complete. The ratio is 20 / sin 63° = 20 / 0.891007 = 22.4465. From there, a = 22.4465 × sin 42° = 15.0197 in and b = 22.4465 × sin 75° = 21.6817 in. The perimeter is 56.7013 in and the area is 0.5 × 15.0197 × 21.6817 × sin 63° = 145.0787 square inches. The only difference from the AAS case is which angle you pair with the given side; the arithmetic is identical afterwards.

The ambiguous SSA case, explained

SSA is the case that trips students and field crews alike. You know sides a and b plus angle A, which is opposite a but not between the two sides. Swing side a from the end of side b and it may miss the base entirely, touch it once, or cross it twice. The test is the height of the triangle measured from the known angle:

With A acute, compare side a against h and against b:

• a < h: side a is too short to reach the base. No triangle. Algebraically, sin B comes out greater than 1.
• a = h: side a just touches. One right triangle with B = 90°.
• h < a < b: side a crosses the base twice. Two triangles, one with an acute B and one with the obtuse supplement.
• a ≥ b: only the acute solution keeps the angle sum under 180°. One triangle.

If angle A is 90° or larger the rule is simpler: there is one triangle when a > b, and none otherwise, because a triangle can hold only one non-acute angle.

SSA sweep: same b and A, different a

Hold b = 10 and A = 30° fixed, so the height is h = 10 × sin 30° = 5.0000, and slide side a through the four zones. Every row below was solved with the sine rule:

Notice the row at a = 9: the second triangle has an angle C of only 3.75° and a side c of 1.177, a sliver so thin it is easy to dismiss, yet it satisfies the measurements exactly. That is the trap of the ambiguous case, and it is why the calculator prints both solutions instead of quietly picking one.

Worked example 3: SSA with two answers

Take a = 7, b = 10 and A = 30°. The height is h = 10 × sin 30° = 5, and 5 < 7 < 10, so expect two triangles. Rearranging the sine rule, sin B = b × sin A / a = 10 × 0.5 / 7 = 0.714286.

• Triangle 1: B = arcsin(0.714286) = 45.58°, so C = 180° − 30° − 45.58° = 104.42°, and c = 14 × sin C = 13.5592 (carrying the unrounded C = 104.4153°). Area 33.8981.
• Triangle 2: B = 180° − 45.58° = 134.42°, so C = 15.58°, and c = 14 × sin C = 3.7613 (carrying the unrounded C = 15.5847°). Area 9.4032.

Both use the same ratio a / sin A = 7 / 0.5 = 14. Both triangles genuinely have a side of 7, a side of 10, and a 30° angle opposite the 7. Only extra information, a rough sketch, a sanity check on the third side, or knowing in advance that the triangle is acute, tells you which one the problem intends.

Sine values for common angles

Because sin (180° − x) equals sin x, the sine column is a mirror around 90°. That symmetry is precisely what creates the ambiguous case, and seeing it in a table makes the idea concrete:

How to use this calculator

1. Pick the case that matches your data: AAS, ASA or SSA. The hint under each button tells you what order the pieces come in.
2. Label the triangle first. Side a must be opposite angle A, side b opposite angle B, side c opposite angle C. Relabeling on paper before you type is the single best way to avoid a wrong answer.
3. Enter the three values. For AAS you give two angles and the side opposite the first one; for ASA the side goes between the two angles; for SSA you give both sides and the angle opposite side a.
4. Choose a unit label (inches, feet, centimeters, meters, or none) and how many decimals you want. The unit is a label only; the math is unit-free, so any consistent unit works.
5. Read the answer. The headline says how many triangles fit. Below it you get every angle, every side, the area, the perimeter, the circumradius, and a numbered derivation you can copy into homework.

If you switch to SSA and see two result cards, the data really is ambiguous and both cards are correct. If you see a red panel instead, the angles do not sum under 180° or a side is not positive.

Who this is for

• Precalculus and trigonometry students checking homework, especially ambiguous-case problems where the textbook answer key lists two triangles.
• Surveyors and civil crews turning a measured baseline and two instrument angles into distances that cannot be tape-measured directly.
• Carpenters and fabricators laying out non-square framing, hip rafters, stair skirts and gussets where no angle is 90°.
• Machinists and welders setting a fixture from one known length and two angles off a drawing.
• Navigators and pilots resolving a triangle of bearings and legs where one leg and its opposite bearing are known.
• Physics and engineering students resolving force or velocity triangles that are not right triangles.

A real measurement: surveying across a creek

You need the distance from stake A to a tree at C, but a creek blocks the tape. Set a second stake B on your own bank, measure the baseline AB = 120 ft, and read the two angles to the tree: 52° at A and 74° at B. The angle at the tree is C = 180° − 52° − 74° = 54°, and the baseline is opposite it, so this is ASA. The ratio is 120 / sin 54° = 120 / 0.809017 = 148.3282. The distance from B to the tree is 148.3282 × sin 52° = 116.88 ft, and from A to the tree it is 148.3282 × sin 74° = 142.58 ft. The enclosed area is about 6,741 square feet. Two angles and one accessible length replaced a measurement nobody could take.

Key terms

• Oblique triangle: any triangle without a 90° angle. The sine and cosine rules exist because SOH-CAH-TOA does not apply to these.
• Included side or angle: the piece that sits between the two other known pieces. ASA has an included side; SAS has an included angle.
• Supplement: 180° minus an angle. Supplements share the same sine, which is the root of the SSA ambiguity.
• Circumradius (R): the radius of the circle through all three vertices. The shared sine-rule ratio equals 2R.
• Height h: in the SSA test, h = b x sin A, the shortest possible distance from the far vertex to the base line. It decides whether side a can reach.
• Degenerate triangle: a "triangle" whose angles sum to exactly 180° with one of them at zero, so it collapses to a straight line. The calculator rejects these.

What changes the result the most

• Mislabeling a side: pairing side b with angle A is the most common error and it silently produces a plausible but wrong triangle.
• Angle precision near 90°: sine changes very slowly around 90°, so an angle read as 88° instead of 90° barely moves the sine. The opposite is true near 0°, where a small angle error swings the answer hard.
• How close a is to h in SSA: right at the boundary the two candidate triangles are nearly identical, and a fraction of a degree flips the answer between two solutions, one, or none.
• Rounding intermediate values: feeding a rounded angle back into the ratio compounds error. Carry full precision until the final display.
• Unit consistency: mixing feet and inches in one triangle scales one side wrongly by a factor of 12 and quietly invalidates everything downstream.

Practical tips

• Sketch first, even roughly. A quick drawing usually shows immediately whether the obtuse SSA solution is physically sensible for your problem.
• Use the biggest-side rule as a check: the largest angle always faces the longest side. If your solution violates that, a label is wrong.
• Solve for the smaller angle first in SSA when you can pick. An angle opposite a shorter side must be acute, which removes the ambiguity outright.
• Cross-check the area two ways. Compute 1/2 x a x b x sin C and then Heron's formula on the three solved sides; matching values confirm the whole solution.
• Verify the ratio. Divide each finished side by the sine of its own angle. All three must agree, as in the table above.

Limitations and assumptions

• It assumes a plane (Euclidean) triangle with angles summing to exactly 180°. Long-baseline geodetic work on the curved earth needs spherical trigonometry instead.
• It needs a complete side-angle pair. SSS and SAS data must go through the law of cosines first.
• Results are rounded for display at the decimal setting you choose; the internal arithmetic keeps full double precision.
• It gives decimals, not exact radicals. An answer of 8.660 will not be shown as 5 times the square root of 3.
• It does not know your measurement error. Field angles carry instrument tolerance, and near the SSA boundary that tolerance can change the number of valid triangles.

Law of sines or law of cosines?

Both rules solve oblique triangles; the difference is what they need to start. The sine rule pairs a side with its opposite angle and is a single multiplication or division per step, so it is faster whenever such a pair exists (AAS, ASA, SSA). The cosine rule, written c² = a² + b² − 2ab x cos C, connects all three sides to one angle and is the only way in when you have SSS or SAS. It also has no ambiguity, because cosine is negative for obtuse angles and positive for acute ones, so an inverse cosine returns a unique answer between 0° and 180°. A practical workflow: start with the cosine rule if you must, use it to find the largest angle first, then finish with the sine rule, whose remaining angles are then guaranteed acute.

How it compares to related calculators

• For SSS or SAS data, or for a general triangle solve with sides and angles mixed, use the Triangle Calculator.
• For sin, cos and tan on their own, degree and radian conversion, or a right-triangle solve, use the Trigonometry Calculator.
• For a missing side when one angle is exactly 90°, use the Pythagorean Theorem Calculator.
• For the area of a triangle from a base and height, or for other shapes, use the Area Calculator, and for the distance around it the Perimeter Calculator.
• For arbitrary evaluation of the inverse sine or any other function while you work, use the Scientific Calculator.

Sources

The law of sines, the 180° angle sum of a plane triangle, the area formula 1/2 x a x b x sin C, Heron's formula and the circumradius relation are exact results of Euclidean geometry. They are deterministic mathematics with no measured or jurisdictional inputs, so no external data source applies, and every figure on this page was computed directly from those definitions rather than quoted from anywhere.

• Degree measure follows the standard convention of 360 degrees to a full turn, with a plane triangle summing to 180 degrees.
• Length labels (inch, foot, centimeter, meter) follow the exact definitions in NIST Special Publication 811, Guide for the Use of the International System of Units (SI), where 1 inch = 25.4 mm exactly. The sine rule itself is unit-free, so any consistent unit gives the same angles.