Law of Cosines Calculator

Real-time triangle side and angle calculator with step-by-step solutions

Triangle Parameters
Calculation Results
Side A
0.00
Opposite angle A
Side B
0.00
Opposite angle B
Side C
0.00
Opposite angle C
Angle A
0.00°
Opposite side A
Angle B
0.00°
Opposite side B
Angle C
0.00°
Opposite side C
Triangle Type

Unknown

Enter values to determine triangle type
Perimeter
0.00
Area
0.00
Law of Cosines Formula Used:
c² = a² + b² - 2ab cos(C)
Triangle Visualization
Calculation History

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Understanding the Law of Cosines: A Comprehensive Guide

What is the Law of Cosines?

The Law of Cosines is a fundamental theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It's an extension of the Pythagorean theorem that works for all types of triangles, not just right triangles.

The Law of Cosines Formula

The standard formula is: c² = a² + b² - 2ab cos(C)

Where:

  • a, b, c are the lengths of the sides of the triangle
  • C is the angle opposite side c

How to Use This Calculator

Our Law of Cosines calculator simplifies complex trigonometric calculations. Here's how to use it effectively:

  1. Enter known values: Input any three parameters of your triangle (sides or angles).
  2. Ensure validity: The calculator automatically checks if your inputs can form a valid triangle.
  3. Get instant results: All remaining sides, angles, perimeter, and area are calculated in real-time.
  4. Visualize: View a dynamic diagram of your triangle with accurate proportions.

Practical Applications

The Law of Cosines calculator is essential for:

  • Engineering: Structural analysis and design
  • Navigation: Calculating distances and bearings
  • Physics: Solving vector problems and force analysis
  • Computer Graphics: 3D modeling and rendering
  • Surveying: Land measurement and mapping
Pro Tips
  • Always include at least one side length for accurate calculations
  • The sum of angles in any triangle is always 180°
  • For right triangles, the Law of Cosines simplifies to the Pythagorean theorem
  • Save frequent calculations for easy reference
  • Use the visualization to verify your results
Key Features
  • Real-time calculations: Instant results as you type
  • Step-by-step solutions: Understand how results are derived
  • Visual representation: See your triangle drawn to scale
  • Calculation history: Track all your previous calculations
  • Multiple export options: Share, print, or save your results
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