Law of Cosines Calculator – Solve Triangle Sides & Angles

Law of Cosines Triangle Solver

Solve triangle sides and angles using the Law of Cosines (SAS & SSS cases).

Not every triangle problem can be solved using the Pythagorean Theorem. That formula only works for right-angled triangles. In real-world geometry, many triangles do not have a 90-degree angle. This is where the Law of Cosines becomes extremely useful.

The Law of Cosines Calculator helps you find missing sides or angles of any triangle, even when it is not a right triangle. By entering a few known values, the calculator instantly gives you accurate results without long calculations.

This tool is perfect for students, teachers, engineers, surveyors, and anyone who needs fast and reliable triangle solutions.

What the Law of Cosines Calculator Is

A Non-Right Triangle Solver

The Law of Cosines Calculator is an online trigonometry tool designed to solve any triangle, not just right triangles. It works with two main cases:

  • SAS (Side-Angle-Side): Two sides and the included angle are known.

  • SSS (Side-Side-Side): All three sides are known.

Using these inputs, the calculator can find:

  • A missing side length

  • A missing angle

What the Calculator Can Find

The calculator provides:

  • Unknown side lengths

  • Unknown angles

  • Validity of the triangle

Some versions also show calculation steps for better understanding.

How the Law of Cosines Calculator Works

Step 1: Enter Known Values

You start by entering the values you already know, such as:

  • Two sides and the included angle

  • Or all three sides

The calculator checks whether the values form a valid triangle.

Step 2: Apply the Law of Cosines

The tool uses the Law of Cosines formula to relate the sides and angles of the triangle.

Step 3: Display the Result

The missing side or angle is calculated instantly and shown on the screen.

Key Formulas Used

Law of Cosines for Finding a Side

If sides a and b and angle C are known:

c^2 = a^2 + b^2 - 2ab\cos(C)

Law of Cosines for Finding an Angle

If all three sides are known:

\cos(C) = \frac{a^2 + b^2 - c^2}{2ab} C = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right)

These formulas allow you to solve any triangle.

Step-by-Step Examples

Example 1: Finding a Missing Side (SAS)

Given:

  • a = 7 units

  • b = 10 units

  • C = 60°

Apply the formula:

c^2 = 7^2 + 10^2 - 2(7)(10)\cos(60^\circ) c^2 = 49 + 100 - 140(0.5) c^2 = 79 c = \sqrt{79} \approx 8.89

So, side c ≈ 8.89 units.

Example 2: Finding a Missing Angle (SSS)

Given:

  • a = 6 units

  • b = 8 units

  • c = 10 units

Apply the formula:

\cos(C) = \frac{6^2 + 8^2 - 10^2}{2(6)(8)} \cos(C) = \frac{36 + 64 - 100}{96} = 0 C = \cos^{-1}(0) = 90^\circ

So, angle C = 90°.

Features of the Law of Cosines Calculator

Works for All Triangle Types

The calculator works for acute, obtuse, and right triangles, making it very flexible.

Instant Results

You get answers in seconds without doing manual trigonometry.

Simple Input System

Only a few values are needed to solve the triangle.

High Accuracy

All calculations follow standard trigonometric rules.

Uses and Applications

Education and Homework

Students use the calculator to solve complex triangle problems and check their answers.

Engineering and Architecture

Engineers use triangle calculations for structural design, layouts, and measurements.

Surveying and Navigation

Surveyors and navigators use the Law of Cosines to calculate distances and angles over land and sea.

Tips to Avoid Common Mistakes

Many users forget that the Law of Cosines is mainly used for non-right triangles. If the triangle has a 90° angle, the Pythagorean Theorem is usually simpler. Always check the triangle type before choosing the formula.

Another common mistake is using the wrong angle. The Law of Cosines requires the included angle between the two known sides. If the angle does not sit between those sides, the result will be incorrect.

Some users mix up degrees and radians. The calculator expects angles in degrees. Entering radians can cause wrong answers, so always confirm the unit.

Incorrect side labeling can also lead to errors. Make sure side c is always opposite angle C, and so on.

Finally, rounding numbers too early can reduce accuracy. Let the calculator handle full precision before rounding.

FAQs

What is the Law of Cosines?

It is a formula that relates the sides and angles of any triangle.

Can it solve right triangles?

Yes, but the Pythagorean Theorem is usually simpler for right triangles.

Is this calculator free?

Most Law of Cosines calculators are free to use online.

Does it work for obtuse triangles?

Yes, it works for all triangle types.

Final Words

The Law of Cosines Calculator is a powerful tool for solving triangle problems that the Pythagorean Theorem cannot handle.

Whether you are studying trigonometry, working in engineering, or solving real-world geometry problems, this calculator saves time, improves accuracy, and makes triangle calculations easy.

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