Laws of Sines & Cosines

Solve non-right triangles. The Law of Sines maps side-to-angle ratios to the circumcircle diameter ($2R$). The Law of Cosines generalizes the Pythagorean Theorem to arbitrary angles.

Interactive Sandbox

Draggable Triangle Board

Drag vertices **A** (blue), **B** (green), or **C** (purple) to reshape the triangle. Observe how side lengths, angle arcs, and ratios adapt in real-time.

Interactive Plane: Drag Vertices A, B, or C

Law of Sines Equation

The dashed circle is the circumcircle. The ratio of side/sin(Angle) is always equal to its diameter.

Algebraic Solver

Step-by-Step Solver

Trace how we compute missing properties using the active vertex coordinates.

Solving Sides & Angles

Active Triangle Summary

Side Lengths

a = 0.00
b = 0.00
c = 0.00

Vertex Angles

A = 0.0°
B = 0.0°
C = 0.0°

Evaluation

Conceptual Quiz

Test your understanding of Sines & Cosines laws, circumcircle diameter relations, and triangle cases.

Question 1 of 5 Score: 0/0

When is it appropriate to use the Law of Cosines rather than the Law of Sines?

Correct Answer!

Explanation text...

Solving Cases Rules

Law of Sines: SAA, ASA, SSA (ambiguous case)
Law of Cosines: SAS, SSS
Circumcircle diameter: 2R = a / sin(A)
Cos C = (a² + b² - c²) / 2ab