Law of Sines Calculator
Solve triangles using the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C).
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
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.
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.
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.
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.
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.
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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
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°.
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
Used in These Calculators
Calculators that build on or apply the concepts from this page:
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.
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