MathIsimple
Back to Calculators
Sine Law Calculator

Sine Law Calculator

Solve triangles using the Law of Sines. Calculate unknown sides and angles with automatic ambiguous case detection and step-by-step solutions.

Geometry Spot/Calculators/Sine Law

Sine Law Calculator

Solve triangles using the sine law. Calculate unknown sides and angles with step-by-step solutions.

Learn Sine Law
Sine Law Calculator
Enter known sides and angles to solve the triangle
Current Setup:
Triangle ABC: a=?, b=?, c=?, A=?°, B=?°, C=?°
Side a is opposite angle A, side b is opposite angle B, side c is opposite angle C
Sine Law Formula
asin⁡A=bsin⁡B=csin⁡C=2R\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R
When to use:
  • AAS: Two angles and one non-included side
  • ASA: Two angles and the included side
  • SSA: Two sides and one angle opposite (ambiguous case)
Ambiguous Case (SSA): May have 0, 1, or 2 solutions
Related Resources
Sine Law Applications

Calculator Features

AAS & ASA Cases

Two angles and one side - always unique solution

asin⁡A=bsin⁡B\frac{a}{\sin A} = \frac{b}{\sin B}

SSA Ambiguous Case

Two sides and opposite angle - may have 0, 1, or 2 solutions

sin⁡B=b⋅sin⁡Aa\sin B = \frac{b \cdot \sin A}{a}

Circumradius

Find the circumscribed circle radius

2R=asin⁡A2R = \frac{a}{\sin A}

The Sine Law Formula

Standard Form

asin⁡A=bsin⁡B=csin⁡C=2R\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Where R is the circumradius of the triangle

When to Use Sine Law

  • ✓AAS: Two angles and a non-included side
  • ✓ASA: Two angles and the included side
  • !SSA: Two sides and an opposite angle (ambiguous)

Understanding the Ambiguous Case (SSA)

No Solution

When sin B > 1

bsin⁡Aa>1\frac{b \sin A}{a} > 1

One Solution

Right triangle or A ≥ 90°

sin B = 1 or a ≥ b

Two Solutions

When A < 90° and a < b

B₁ = arcsin(x), B₂ = 180° - B₁

Related Tools & Learning

Sine Law Lesson

Learn the theory and applications of sine law

Cosine Law Calculator

For SAS and SSS triangle problems

Triangle Area Calculator

Calculate triangle area using multiple methods

Frequently Asked Questions

The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C) = 2R, where a, b, c are sides, A, B, C are opposite angles, and R is the circumradius of the triangle.
Ask AI ✨