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Calculator Answers

Law of Sines Calculator

Blake Boege
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.

See also: Sin Cos Tan Calculator, Law of Cosines Calculator, and Right Triangle Calculator.

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.

e.g. 35

e.g. 65

e.g. 12

ABCabc
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° − A − B, then solve sides via sine ratios.
  • ASA (Angle-Side-Angle): Subtract the two angles from 180° 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 · sin(A) < a < b.

Related Calculators

More tools from Education

Sin Cos Tan CalculatorCalculate sine, cosine, and tangent values for any angle in degrees or radians. Shows reciprocal functions csc, sec, and cot.Law of Cosines CalculatorSolve any triangle using c² = a² + b² − 2ab·cos(C) with SAS and SSS cases.Right Triangle CalculatorSolve a right triangle from any of two legs, leg + hypotenuse, leg + acute angle, or hypotenuse + acute angle. Returns all sides, both acute angles, area, and perimeter.Reference Angle CalculatorFind the positive acute reference angle for any angle in degrees or radians (including fractions of pi) with a unit circle visualizer and step-by-step rules.

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.