Law of Cosines Calculator
Calculate triangle sides or angles and view step-by-step solutions.
The Law of Cosines Calculator helps you calculate missing sides or angles of any triangle quickly and accurately. This tool is especially useful when dealing with non-right triangles, where basic trigonometric rules like sine or cosine ratios alone are not sufficient.
Whether you are a student, teacher, engineer, architect, or math enthusiast, this calculator provides instant results along with step-by-step explanations, making it ideal for both learning and professional use.
What Is the Law of Cosines?
The Law of Cosines is a fundamental trigonometric formula that generalizes the Pythagorean theorem for all types of triangles. It relates the lengths of the sides of a triangle to the cosine of one of its angles.
Law of Cosines Formula
For a triangle with sides a, b, and c, and angle C opposite side c:
- c² = a² + b² − 2ab cos(C)
This formula allows you to:
- Find a missing side when two sides and the included angle are known
- Find a missing angle when all three sides are known
When Should You Use the Law of Cosines?
You should use the Law of Cosines when working with:
- Scalene triangles
- Obtuse triangles
- Acute triangles
- Problems where the triangle is not right-angled
Common real-world and academic applications include:
- Geometry and trigonometry problems
- Physics and engineering calculations
- Surveying and land measurement
- Construction and architectural planning
- Navigation and vector analysis
How to Use the Law of Cosines Calculator
Using this math calculator is simple and intuitive:
- Enter any three known values (two sides and one angle, or three sides)
- Leave the unknown value empty
- Click Calculate
- Instantly view:
- The computed side or angle
- A clear step-by-step breakdown of the calculation
The calculator automatically determines whether to compute a side length or an angle, eliminating guesswork.
Example Calculation
Example 1: Finding a Side
If:
- Side a = 7
- Side b = 9
- Angle C = 45°
The calculator applies the formula:
c² = 7² + 9² − 2(7)(9)cos(45°)
Result:
- Side c ≈ 6.72 units
Example 2: Finding an Angle
If:
- Side a = 5
- Side b = 8
- Side c = 10
The calculator rearranges the formula to find the angle:
cos(C) = (a² + b² − c²) / 2ab
Result:
- Angle C ≈ 86.18°
Why Use This Law of Cosines Calculator?
This calculator is designed for accuracy, clarity, and usability:
- ✔ Fully mobile responsive
- ✔ Desktop full-width layout
- ✔ Instant results
- ✔ Step-by-step explanations
- ✔ No ads or distractions
- ✔ Suitable for students and professionals
Unlike basic calculators, this tool focuses on understanding, not just answers.
Law of Cosines vs Law of Sines
| Feature | Law of Cosines | Law of Sines |
|---|---|---|
| Works for non-right triangles | Yes | Yes |
| Requires angle-side-side data | Yes | Limited |
| Best for obtuse triangles | Yes | No |
| Used to find angles from sides | Yes | Sometimes |
The Law of Cosines is generally preferred when two sides and the included angle or three sides are known.
Frequently Asked Questions
Is the Law of Cosines the same as the Pythagorean theorem?
No. The Pythagorean theorem is a special case of the Law of Cosines where the angle is 90°.
Can this calculator handle obtuse triangles?
Yes. The Law of Cosines works for acute, right, and obtuse triangles.
Are angles calculated in degrees?
Yes. This calculator uses degrees, which is standard for most educational and real-world applications.
Final Notes
The Law of Cosines Calculator is a reliable tool for solving complex triangle problems efficiently. With its clean design, responsive layout, and educational explanations, it serves both practical calculations and learning purposes.
For fast, accurate, and understandable triangle solutions, this calculator delivers everything you need.