Law of Cosines Calculator

Category: Algebra II
- May 14, 2025
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Calculate missing sides or angles in a triangle using the Law of Cosines. This calculator helps solve triangles when you know three pieces of information about a triangle.

Triangle Input

a b c A B C
units
units
units
°
°
°

Display Options

About the Law of Cosines

The Law of Cosines is a fundamental formula in trigonometry that relates the sides and angles of a triangle. It's a generalization of the Pythagorean theorem to any triangle, not just right triangles.

When to Use the Law of Cosines
  • SAS (Side-Angle-Side): When you know two sides and the included angle
  • SSS (Side-Side-Side): When you know all three sides
  • SSA (Side-Side-Angle): When you know two sides and a non-included angle (may have 0, 1, or 2 solutions)
Applications

The Law of Cosines is used in various fields including:

  • Surveying and navigation
  • Engineering and construction
  • Physics and vector analysis
  • Astronomy for calculating distances between celestial objects
  • Computer graphics for 3D modeling

Law of Cosines Formulas:

c² = a² + b² - 2ab·cos(C)

b² = a² + c² - 2ac·cos(B)

a² = b² + c² - 2bc·cos(A)

Finding Angles:

cos(C) = (a² + b² - c²) / (2ab)

cos(B) = (a² + c² - b²) / (2ac)

cos(A) = (b² + c² - a²) / (2bc)

What is the Law of Cosines Calculator?

The Law of Cosines Calculator is a simple and efficient tool that helps you solve triangles by finding unknown sides or angles. Whether you know two sides and the included angle (SAS), all three sides (SSS), or are working with side-side-angle (SSA) configurations, this calculator can quickly complete the triangle for you. It’s especially helpful in situations where the basic Pythagorean theorem doesn't apply.

How the Calculator Helps You

This calculator saves time and reduces error when solving for missing triangle parts. It can also show step-by-step solutions, making it easier to follow the logic behind the calculations. Whether you're handling school assignments, construction designs, or surveying tasks, having a reliable Law of Cosines Calculator is incredibly useful.

  • Quickly calculate missing sides or angles
  • Works for SAS, SSS, and SSA cases
  • Shows clear, detailed calculation steps
  • Offers visual diagrams of your triangle
  • Provides additional information like area and perimeter

How to Use the Law of Cosines Calculator

  • Choose what you want to find: a missing side or an angle.
  • Enter the known sides and angles into the calculator.
  • Select the unit for angles: degrees or radians.
  • Choose how many decimal places you want in your results.
  • Click the "Calculate" button to see the solution.
  • Review the triangle visualization and solution steps if needed.

Where the Law of Cosines is Useful

Understanding the Law of Cosines is valuable in various fields:

  • Surveying: Calculating land boundaries and distances.
  • Navigation: Solving triangles for mapmaking and travel paths.
  • Engineering: Designing structures and analyzing forces.
  • Astronomy: Measuring distances between stars or planets.
  • Computer Graphics: Building realistic 3D models.

Frequently Asked Questions (FAQ)

What does the Law of Cosines Calculator solve?

It solves triangles by finding missing sides or angles using known values, especially when standard right-triangle formulas do not work.

Can I use it for any triangle?

Yes, it works for any triangle, not just right triangles.

Do I need all three sides to use it?

No. You can use it if you have two sides and an included angle (SAS) or all three sides (SSS). The tool also handles SSA cases for angle finding.

Does it show how the answer was found?

Yes! The calculator can display detailed step-by-step calculations so you can follow the Math easily.

Is there anything similar that might help?

Yes, depending on your math needs, you might also find these calculators useful:

Final Thoughts

The Law of Cosines Calculator is a straightforward and reliable way to solve triangle problems without manual work. It not only speeds up calculations but also helps reinforce your understanding of triangle relationships. For those exploring additional topics like calculating inverse sinh with an Inverse Hyperbolic Sine Calculator or solving polynomial roots with a Polynomial Roots Calculator, a strong foundation in trigonometric solutions like this one can be a big advantage.