Law of Cosines Triangle Calculator

Calculate Law of Cosines Triangle

Enter any two sides and the included angle, or all three sides, and the Law of Cosines Triangle Calculator will instantly solve for missing angles, sides, and area of the triangle.

Solve a Triangle

Enter the three values you know to find the rest.

Side a:

Side b:

Side c:

Angle A (°):

Angle B (°):

Angle C (°):

Related Calculators

Law of Cosines Calculator Law of Cosines Triangle Calculator Cosine Similarity Calculator Cosine Distance Calculator Fourier Cosine Series Calculator Hyperbolic Cosine Calculator

Law of Cosines Formula

The **Cosine Rule** is used in triangles when you know two sides and the included angle (**SAS**) or all three sides (**SSS**).

To find a side:

$$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$ $$a^2 = b^2 + c^2 - 2bc \cdot \cos(A)$$ $$b^2 = a^2 + c^2 - 2ac \cdot \cos(B)$$

To find an angle:

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

Example Table

Input Values	Formula Used	Output
a = 7, b = 9, C = 120°	$$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$	c = 14.42
a = 5, b = 6, c = 7	$$\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$$	C = 74.5°

How It Works

This calculator automatically detects whether you've provided a **SAS** (two sides & an included angle) or **SSS** (three sides) triangle and applies the correct formulas.

If you know two sides and the included angle, the calculator finds the third side and the remaining angles using the Law of Sines.
If you know three sides, the calculator finds all three angles.
In both cases, it calculates the triangle's area.

User Guide

Enter the three values you know into the corresponding input fields. Make sure to only fill in three boxes.
Press the "Calculate" button.
View the results for the missing side(s), angle(s), and area.

Frequently Asked Questions (FAQs)

Q1. What is the Law of Cosines Calculator used for?

It helps solve **oblique triangles** and **non-right triangles** when you know two sides and the included angle or all three sides.

Q2. Can I use this Triangle Calculator for right triangles?

Yes, but the Pythagorean theorem is often simpler. This tool works for any triangle in **geometry problems**.

Q3. What’s the difference between Law of Cosines and Cosine Similarity Calculator?

The **Law of Cosines Calculator** is for triangles in geometry. The **Cosine Similarity Calculator** is used for comparing vectors and datasets in machine learning.

Q4. Does the calculator show answers in degrees or radians?

Yes, the results for angles are displayed in both degrees and radians.