Law of Sines Calculator

Q: What is the law of sines?

a / sin A = b / sin B = c / sin C. Each side over the sine of the opposite angle is the same ratio, equal to the circumdiameter 2R.

Q: What is the ambiguous case?

SSA: two sides and a non-included angle. Depending on a, b, and A there may be zero, one, or two triangles. The extra candidate is B′ = 180° − arcsin(b sin A / a).

Q: When do you use sines instead of cosines?

Use sines for AAS, ASA, and SSA. Use cosines for SAS and SSS.

Q: Can both SSA triangles be acute?

No. The two B values are supplementary, so one is acute and one is obtuse (unless B = 90°).

Q: Does this solve a right triangle?

It can, but the Right Triangle Calculator uses SOH CAH TOA directly. Use this page when no angle is given as 90°, or when you are studying the ambiguous case.

Q: Are my numbers sent to a server?

No. The calculation runs locally in your browser.

Solve a triangle from two angles and a side (AAS / ASA), or from two sides and a non-included angle (SSA). SSA can produce no triangle, one triangle, or two triangles — the ambiguous case.

Quick Answera / sin A = b / sin B = c / sin C. Angles 40° and 60° with opposite side a = 8 give C = 80°, b ≈ 10.778, c ≈ 12.257. SSA with a = 8, A = 40°, b = 10 has two triangles.

a / sin A = b / sin B = c / sin C

Result

b = 10.778462017

C = 80°, c = 12.25655931

Angle B60°

Angle C80°

Side c12.25655931

Area33.62135387

KindAAS / ASA

Steps

C = 180° − 40° − 60° = 80°.
b = 8 × sin 60° / sin 40° ≈ 10.778.
c = 8 × sin 80° / sin 40° ≈ 12.257.

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How to Use the Law of Sines Calculator

AAS / ASA needs two angles and any one side. The third angle is 180° minus the other two, then each unknown side is the known side times a sine ratio. SSA needs two sides and the angle opposite one of them. That case can be empty, unique, or ambiguous.

Choose AAS / ASA or SSA.
Enter the two angles and a side, or side a, angle A, and side b. A is opposite a.
Read the remaining parts. If SSA has two solutions, the second triangle appears under the first.

Two sides and the included angle, or three sides, belong on the Law of Cosines Calculator. A right triangle with SOH CAH TOA belongs on the Right Triangle Calculator.

The Law of Sines

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

Each side is proportional to the sine of the opposite angle. The common ratio equals the diameter of the circumcircle. Once two angles are known, the third is determined because the angles of a triangle sum to 180°. Then every side follows from one known side.

The identity does not, by itself, tell you which of two supplementary angles is B when you only know sin B. That is why SSA is called ambiguous: arcsin gives an acute B, and 180° − B may also fit.

The Ambiguous Case (SSA)

Given a, A, and b, drop a perpendicular from the unknown vertex to side b. Its length is h = b sin A.

If A is acute and a < h, the side is too short: no triangle.
If A is acute and a = h, the triangle is right: one solution.
If A is acute and h < a < b, two triangles fit the same data.
If A is acute and a ≥ b, only the acute-B triangle fits.
If A is right or obtuse, you need a > b or there is no triangle.

The calculator reports both triangles when both angle sums stay under 180°. It does not pick a “preferred” one — homework figures usually include a sketch that selects acute B or obtuse B.

Law of Sines Examples

AAS: A = 40°, B = 60°, a = 8

C = 80°. b = 8 sin 60° / sin 40° ≈ 10.778. c = 8 sin 80° / sin 40° ≈ 12.257.

AAS: 30°, 45°, a = 10

C = 105°. The triangle is obtuse at C. b = 10 sin 45° / sin 30° ≈ 14.142. c = 10 sin 105° / sin 30° ≈ 19.319.

SSA two triangles: a = 8, A = 40°, b = 10

h = 10 sin 40° ≈ 6.428. Because h < 8 < 10, two triangles exist. B ≈ 53.46° or 126.54°.

SSA one triangle (height match): a = 5, A = 30°, b = 10

h = 10 sin 30° = 5 = a, so the triangle is right at B. One solution: B = 90°, C = 60°, c = 5√3 ≈ 8.660.

SSA one triangle (a longer than b): a = 12, A = 40°, b = 10

a > b, so only the acute-B triangle fits. The supplementary B would make A + B exceed 180°.

SSA no triangle: a = 4, A = 30°, b = 10

h = 5 and a = 4 < h. Side a cannot reach the other ray. sin B would be greater than 1.

Common Law of Sines Mistakes

Pairing a side with the wrong angle

a must be opposite A. If the given angle sits between the two given sides, that is SAS — use the law of cosines, not this page.

Ignoring the second SSA triangle

arcsin returns an acute angle. Always test 180° − B. If A + (180° − B) is still less than 180°, both triangles are valid.

Letting two angles add to 180° or more

Then C is not a positive interior angle. The data do not describe a triangle.

Using sines for SSS or SAS

Three sides, or two sides and the included angle, need the law of cosines first. Sines cannot start without an opposite pair.

Treating a right triangle as SSA homework

If one angle is 90°, SOH CAH TOA on the Right Triangle Calculator is shorter and avoids the ambiguous-case checklist.