law sines calculator - Aaron Graves, PhDude Replica

Interactive Law of Sines Calculator

Use this calculator with one known side-angle opposite pair and one additional value. You can solve for a missing side or a missing angle, including ambiguous SSA cases.

Known side (for example, a)

Known opposite angle in degrees (for example, A)

Target side (optional, for solving a missing angle)

Target angle in degrees (optional, for solving a missing side)

Fill at least one target value. If you enter both, the calculator checks consistency.

Formula used: a/sin(A) = b/sin(B) = c/sin(C). Angles should be entered in degrees.

What is the Law of Sines?

The Law of Sines is a triangle relationship that connects side lengths with the sines of their opposite angles. If a triangle has sides a, b, c opposite angles A, B, C, then:

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

This rule is especially helpful for non-right triangles, where basic right-triangle trig doesn’t apply directly. It allows you to solve missing sides or angles when enough information is known.

When to Use This Calculator

Best Use Cases

ASA (Angle-Side-Angle): two angles and one side known.
AAS (Angle-Angle-Side): two angles and one non-included side known.
SSA (Side-Side-Angle): two sides and one non-included angle known, which may have 0, 1, or 2 solutions.

The calculator above is designed around a known side-angle pair plus one additional value. It will automatically detect ambiguous SSA possibilities and list all valid triangles.

How to Use the Law of Sines Calculator

To Find a Missing Side

Enter the known side and its opposite angle.
Enter the target opposite angle.
Click Calculate.

The tool computes the missing side, then also gives the third angle and third side to complete the triangle.

To Find a Missing Angle

Enter the known side and its opposite angle.
Enter another side as the target side.
Leave target angle blank and click Calculate.

For SSA inputs, the calculator may return two valid angle results. That is normal and mathematically correct.

Understanding the Ambiguous Case (SSA)

The ambiguous case happens because the inverse sine function can produce two angles between 0° and 180° that share the same sine value: one acute and one obtuse. Both can form valid triangles if each keeps the total angle sum below 180°.

No solution: geometry is impossible with entered values.
One solution: only one angle passes the triangle-angle-sum check.
Two solutions: both acute and obtuse options are valid.

Common Mistakes to Avoid

Mixing angle and side that are not opposite each other for the known pair.
Entering angles in radians when the calculator expects degrees.
Forgetting that all triangle angles must sum to exactly 180°.
Assuming SSA always gives one answer.

Practical Applications

The Law of Sines appears in surveying, navigation, architecture, and engineering design. Anytime you have line-of-sight distances with measured angles, this relationship can quickly recover unknown sides or directions.

In classroom settings, it’s also a foundational bridge from basic trigonometry to broader geometric modeling. A reliable calculator helps you verify hand work and build intuition faster.

Quick Reference

Core Formula