Law of Cosines Calculator
Enter triangle sides and/or angles to calculate missing values using the Law of Cosines.
This Calculator will let users input triangle sides and/or angles, then apply the Law of Cosines:
and its variations for other sides/angles.
Formulas for All Angles
Using the Law of Cosines:
- $$A = \cos^{-1}\left(\frac{b^2 + c^2 – a^2}{2bc}\right)$$
- $$B = \cos^{-1}\left(\frac{a^2 + c^2 – b^2}{2ac}\right)$$
- $$C = \cos^{-1}\left(\frac{a^2 + b^2 – c^2}{2ab}\right)$$
Law of Cosines Calculator – Solve Any Triangle with Ease
The Law of Cosines Calculator is a powerful trigonometry tool that helps you solve triangles by calculating unknown side lengths or angles. It is especially useful for non-right triangles, where the Pythagorean theorem cannot be applied.
With just a few known values, this calculator instantly delivers accurate results, making triangle problems quick and stress-free.
What Is the Law of Cosines?
The Law of Cosines is a trigonometric formula that relates the sides and angles of any triangle.
It states:
c² = a² + b² − 2ab cos(C)
Where:
a, b, c = sides of the triangle
C = angle opposite side c
There are also two other versions:
a² = b² + c² − 2bc cos(A)
b² = a² + c² − 2ac cos(B)
This formula is a generalization of the Pythagorean theorem and works for all triangles.
What the Law of Cosines Calculator Does
This calculator allows you to:
✔ Find unknown side lengths
✔ Calculate missing angles
✔ Solve oblique triangles
✔ Get fast and accurate results
It handles all trigonometric operations automatically.
When Should You Use the Law of Cosines?
The Law of Cosines is ideal when you know:
Case 1: Two sides and the included angle (SAS)
You can directly compute the third side.
Case 2: All three sides (SSS)
You can find any unknown angle.
How to Use the Law of Cosines Calculator
- Enter the known side lengths
- Enter the known angle (if applicable)
- Click Calculate
The calculator will instantly display:
• Missing sides
• Missing angles
• Completed triangle measurements
Why Use an Online Law of Cosines Calculator?
Saves Time
Avoid manual formula rearrangement.
Ensures Accuracy
Eliminates trigonometric errors.
Beginner Friendly
Perfect for students and learners.
Accessible Anywhere
Works on any device.
Example Calculation
Suppose:
Side a = 5 units
Side b = 7 units
Angle C = 60°
Using the formula:
c² = 5² + 7² − 2(5)(7)cos(60°)
c² = 25 + 49 − 70(0.5)
c² = 39
c ≈ 6.24 units
The calculator computes this instantly.
Practical Applications of the Law of Cosines
Geometry and Trigonometry
Solving triangle problems in school and college.
Construction and Surveying
Measuring distances and angles.
Engineering
Analyzing forces and structures.
Navigation
Determining unknown distances.
Advantages of Using Our Law of Cosines Calculator
✔ Fast results
✔ Simple interface
✔ Accurate formulas
✔ No downloads required
✔ Free to use
Law of Cosines vs Law of Sines
| Feature | Law of Cosines | Law of Sines |
|---|---|---|
| Best for | SAS, SSS cases | ASA, AAS, SSA cases |
| Handles right triangles | Yes | Yes |
| Common use | Find sides/angles | Find sides/angles |
Both formulas are important for solving triangles.
Final Thoughts
The Law of Cosines Calculator is an essential trigonometry tool for solving triangles quickly and accurately. Whether you’re dealing with side-angle-side situations or solving with three known sides, this calculator makes complex calculations easy.
By automating the Law of Cosines formula, it saves time, improves accuracy, and helps users understand triangle relationships more clearly.
Try the Law of Cosines Calculator today and simplify your triangle calculations.