Math July 13, 2026 · 8 Min Read

Law of Sines Calculator – Guide & Formulas

Solve any triangle using the Law of Sines. Find missing sides or angles given AAS, ASA, or SSA configurations with step-by-step results and formulas.

The Law of Sines Calculator solves any triangle given two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA). Enter your known values to instantly find all missing sides and angles, with the ambiguous case explained when applicable.

Key Takeaway

Use the free Law of Sines Calculator to solve any triangle using the law of sines. find missing sides or angles given aas, asa, or ssa configurations with step-by-step results and formulas. Get instant results with step-by-step explanations.

How to Use the Law of Sines Calculator

  1. Select the known configuration: AAS, ASA, or SSA.
  2. Enter the known side lengths and/or angles.
  3. The calculator computes all missing sides and angles.
  4. Review the step-by-step solution and check for the ambiguous case.

The Formula

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

Variable Definitions

  • a, b, c: The three sides of the triangle
  • A, B, C: The angles opposite sides a, b, c respectively
  • sin(A): Sine of angle A
  • a/sin(A): The ratio of a side to the sine of its opposite angle (constant for all three pairs)

Example: AAS Triangle with A=40°, B=60°, a=10

Find side b, side c, and angle C given two angles and a non-included side.

  1. Step 1: Find angle C: C = 180° - A - B = 180° - 40° - 60° = 80°.
  2. Step 2: Apply law of sines: b/sin(B) = a/sin(A).
  3. Step 3: Solve for b: b = a·sin(B)/sin(A) = 10·sin(60°)/sin(40°) = 10·0.866/0.643 ≈ 13.47.
  4. Step 4: Apply law of sines for c: c = a·sin(C)/sin(A) = 10·sin(80°)/sin(40°) = 10·0.985/0.643 ≈ 15.32.
  5. Step 5: Verify: angles sum to 180° ✓, ratios are equal ✓.

Frequently Asked Questions

What is the law of sines?

The law of sines states that a/sin(A) = b/sin(B) = c/sin(C) for any triangle. This means the ratio of each side to the sine of its opposite angle is constant. It is useful for solving triangles when you know two angles and a side (AAS or ASA).

When should I use the law of sines?

Use the law of sines when you know: (1) two angles and a non-included side (AAS), (2) two angles and the included side (ASA), or (3) two sides and a non-included angle (SSA). It is not ideal for SSS or SAS cases (use law of cosines instead).

What is the ambiguous case (SSA)?

The ambiguous case occurs when you know two sides and a non-included angle (SSA). This can produce 0, 1, or 2 possible triangles. The law of sines may yield two valid angle solutions (one acute, one obtuse) when the given angle is acute and the opposite side is shorter than the adjacent side but longer than the altitude.

How do I know if there are 0, 1, or 2 solutions in SSA?

Given angle A and sides a (opposite) and b (adjacent): If a < b·sin(A), no triangle exists. If a = b·sin(A), exactly one right triangle exists. If a ≥ b, exactly one triangle exists. If b·sin(A) < a < b, two triangles exist (ambiguous case).

How do I find a side using the law of sines?

Set up the proportion: a/sin(A) = b/sin(B). If you know a, A, and B, solve for b: b = a·sin(B)/sin(A). The key is to have a complete pair (side and its opposite angle) to establish the ratio.

How do I find an angle using the law of sines?

Set up the proportion: a/sin(A) = b/sin(B). If you know a, b, and A, solve for sin(B): sin(B) = b·sin(A)/a. Then take arcsin to find angle B. Be aware of the ambiguous case for SSA.

What is the difference between law of sines and law of cosines?

The law of sines relates sides to their opposite angles (a/sin(A) = b/sin(B)). The law of cosines relates three sides and one angle (c² = a² + b² - 2ab·cos(C)). Use law of sines for AAS, ASA, SSA; use law of cosines for SSS, SAS.

Can the law of sines solve right triangles?

Yes. For a right triangle with C = 90°, sin(C) = 1, so the law of sines becomes a/sin(A) = b/sin(B) = c. This is equivalent to the basic trigonometric ratios (sin = opposite/hypotenuse).

How do I verify my solution is correct?

Check that: (1) all angles sum to 180°, (2) all side/angle ratios are equal (a/sin(A) = b/sin(B) = c/sin(C)), (3) the largest side is opposite the largest angle and the smallest side is opposite the smallest angle.

What are common mistakes with the law of sines?

Common mistakes include: (1) not checking for the ambiguous case in SSA, (2) using the wrong angle-side pairs, (3) forgetting that arcsin can return two possible angles, (4) mixing up degrees and radians in calculations.