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Law of Sines Calculator

Last updated: June 19, 2026

Written by Blake Boege · Founder, Calculator Answers

A Law of Sines calculator solves oblique or right triangles using the trigonometric ratio of side lengths to sines of opposite angles. Given two angles and a side (AAS, ASA) or two sides and a non-included angle (SSA), the calculator uses the formula to find the remaining angles and side lengths. It automatically detects and evaluates the ambiguous case (SSA) where zero, one, or two valid triangles exist.

Solve any triangle using the Law of Sines formulas for AAS, ASA, and SSA cases. Fully supports the ambiguous case with step-by-step math.

Quick Answer

Solve missing sides and angles of a triangle using the Law of Sines. Supports AAS, ASA, and SSA inputs, showing all possible solutions.

Triangle Congruence Case

Angle A (deg)

e.g. 35

Angle B (deg)

e.g. 65

Side a

e.g. 12

Triangle Result

Solved side c

20.6035

Angle C = 80°

Side a12

Side b18.9612

Side c20.6035

Angle A35°

Angle B65°

Angle C80°

Step-by-step solution

Step 1:Find the third angle C: C = 180° - A - B = 180° - 35° - 65° = 80°

Step 2:Find side b using Law of Sines: b = a × sin(B) / sin(A)

Step 3:b = 12 × sin(65°) / sin(35°) = 12 × 0.9063 / 0.5736 = 18.9612

Step 4:Find side c using Law of Sines: c = a × sin(C) / sin(A)

Step 5:c = 12 × sin(80°) / sin(35°) = 12 × 0.9848 / 0.5736 = 20.6035

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Examples

AAS: A = 35°, B = 65°, a = 12

C = 80°, b ≈ 18.96, c ≈ 20.61

ASA: A = 45°, c = 15, B = 60

C = 75°, a ≈ 11.00, b ≈ 13.45

How it works

The Law of Sines Formula

For any triangle with side lengths $a$, $b$, $c$ and opposite angles $A$, $B$, $C$:

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

We can rearrange these proportions to solve for missing sides or angles when enough congruence parameters are specified.

To solve right triangles directly, visit our standard Pythagorean theorem calculator or find 2D area configurations using our triangle area calculator.

AAS, ASA, and SSA Cases

AAS (Angle-Angle-Side): Find the third angle using $180^\circ - A - B$, then solve sides via sine ratios.
ASA (Angle-Side-Angle): Subtract the two angles from $180^\circ$ to find the third angle, then resolve sides.
SSA (Side-Side-Angle): The ambiguous case. Can yield 0, 1, or 2 triangles depending on whether $b \cdot \sin(A) < a < b$.

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Frequently asked questions

The Law of Sines is a trigonometric relationship that states the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides: a/sin(A) = b/sin(B) = c/sin(C).

Use the Law of Sines when you know: (1) two angles and one side (AAS or ASA), or (2) two sides and a non-included angle (SSA). Use the Law of Cosines when you know: (1) two sides and the included angle (SAS), or (2) all three sides (SSS).

The ambiguous case occurs when you are given two sides and a non-included angle (SSA). Depending on the lengths, there may be: (1) no valid triangle, (2) exactly one right or oblique triangle, or (3) two distinct valid triangles (one acute, one obtuse).

No. The Law of Sines is extremely powerful because it applies to *all* triangles, including oblique triangles (acute or obtuse), not just right triangles.

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