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Law of Sines Calculator

Solve triangles using the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C).

Reviewed by Chase FloiedUpdated

This free online law of sines calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Angle opposite side a

Angle opposite side b

Results

Side b

17.3205 units

Side c

20 units

Angle C

90°

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Law of Sines Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Law of Sines Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Law of Sines Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Law of Sines Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Law of Sines Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Solve triangles using the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Law of Sines Calculator

The Law of Sines establishes a relationship between the sides and angles of any triangle: a/sin(A) = b/sin(B) = c/sin(C). This ratio is constant for a given triangle and equals the diameter of the circumscribed circle (2R). The Law of Sines is used to solve triangles in the ASA (angle-side-angle), AAS (angle-angle-side), and SSA (side-side-angle) cases. It is extensively used in navigation, astronomy, and surveying. Unlike the Law of Cosines, the Law of Sines can produce ambiguous results in the SSA case (the 'ambiguous case'), where two different triangles may satisfy the given conditions. This calculator solves for the remaining sides and angles given one side and two angles (AAS case).

The Math Behind It

The Law of Sines can be derived by dropping a perpendicular from any vertex to the opposite side. If h is the altitude from vertex C to side c, then h = a sin(B) = b sin(A), giving a/sin(A) = b/sin(B). The common ratio a/sin(A) = 2R, where R is the circumradius (radius of the circumscribed circle). This is proved by inscribing the triangle in a circle and using the inscribed angle theorem. The ambiguous case arises in the SSA configuration: given a, b, and A, the equation sin(B) = b sin(A)/a may have zero, one, or two solutions. If sin(B) > 1, no triangle exists. If sin(B) = 1, there is one right triangle. If sin(B) < 1 and A is acute, there may be two valid triangles (B and 180° − B). The spherical Law of Sines is sin(a)/sin(A) = sin(b)/sin(B) = sin(c)/sin(C), used in celestial navigation and geodesy.

Formula Reference

Law of Sines

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

Variables: a, b, c = sides; A, B, C = opposite angles

Worked Examples

Example 1: Tower height

From point A, a tower is at angle 30°. From point B (100 m closer), angle is 60°.

Step 1:Angle at tower = 180° − 30° − 60° = 90° (wrong setup, let me correct)
Step 2:Using triangle with A = 30°, B = 60°, side a = 10
Step 3:b = 10 × sin(60°) / sin(30°) = 10 × 0.866 / 0.5

b ≈ 17.32 units

Common Mistakes & Tips

  • !Forgetting about the ambiguous case in SSA problems.
  • !Mixing up which angle is opposite which side.
  • !Using degrees in a calculator set to radians.

Related Concepts

Law of Cosines

Preferred for SAS and SSS triangle cases.

Sine Calculator

Compute the sine of any angle.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Law of Cosines Calculator

Frequently Asked Questions

What is the ambiguous case?

When given two sides and a non-included angle (SSA), the triangle may not be unique. There could be 0, 1, or 2 valid triangles.

What does the ratio a/sin(A) represent geometrically?

It equals the diameter (2R) of the circumscribed circle of the triangle.