Law of Cosines Calculator is a powerful geometric tool for students and engineers. It solves complex triangle problems instantly. This tool helps you find missing side lengths or unknown angles easily. You get highly accurate mathematical results without doing complex trigonometry yourself.
⚡ What is the Law of Cosines
The Law of Cosines is a famous rule in geometry. It connects the lengths of a triangle's sides to the cosine of one of its angles. People use this rule when they cannot use the Pythagorean theorem. It works perfectly for triangles that do not have a standard right angle.
📊 How to Use This Calculator
Follow these easy steps to solve your triangle problems:
🔹 Select your calculation mode from the top dropdown menu.
🔹 Choose SAS mode if you know two sides and the angle between them.
🔹 Choose SSS mode if you know all three side lengths.
🔹 Enter your numerical values in the input boxes.
🔹 The tool instantly calculates the missing side or missing angle.
🔢 Conversion Formula
The math behind this tool uses two standard geometric equations.
Find Side (SAS) Formula:
c² = a² + b² - 2ab × cos(C)
Example: Sides 3 and 4 with a 90 degree angle gives side 5.
Find Angle (SSS) Formula:
cos(C) = (a² + b² - c²) ÷ (2ab)
Example: Sides 3 and 4 and 5 gives a 90 degree angle.
💡 Simple Explanation
Think of this tool as an upgraded version of the Pythagorean theorem. When a triangle leans over and loses its perfect right angle the normal math stops working. The Law of Cosines adds a special adjustment piece to the equation. This adjustment handles the tilt perfectly. Our calculator runs these complex trigonometric functions flawlessly.
📊 Common Triangle Sample Table
| Side a | Side b | Angle C or Side c | Result |
|---|---|---|---|
| 3 | 4 | Angle 90 | Side c = 5 |
| 5 | 12 | Angle 90 | Side c = 13 |
| 5 | 7 | Angle 45 | Side c = 4.9507 |
| 10 | 10 | Angle 60 | Side c = 10 |
| 8 | 11 | Angle 30 | Side c = 5.7061 |
| 3 | 4 | Side c = 5 | Angle C = 90 |
| 5 | 5 | Side c = 5 | Angle C = 60 |
| 7 | 8 | Side c = 9 | Angle C = 73.3984 |