Cosine Rule Calculator
Instantly solve any triangle by finding missing sides or angles using the Law of Cosines. Perfect for trigonometry, physics, and engineering calculations.
Triangle Parameters
Select what you want to calculate and enter the known values
Choose whether you need to solve for a missing side length or an interior angle.
Length of the first known side
Length of the second known side
The angle between side a and side b
Triangle Solution
Law of Cosines breakdown
Enter the known triangle measurements, then click Solve Triangle to calculate the missing side or angle using the Cosine Rule.
Law of Cosines Variations
The Cosine Rule can be rearranged to solve for any side or angle in a triangle.
| Target | Formula | Known Values |
|---|---|---|
| Find Side c | c = √(a² + b² – 2ab cos(C)) | Sides a, b & Angle C |
| Find Side a | a = √(b² + c² – 2bc cos(A)) | Sides b, c & Angle A |
| Find Side b | b = √(a² + c² – 2ac cos(B)) | Sides a, c & Angle B |
| Find Angle C | C = acos((a² + b² – c²) / 2ab) | Sides a, b, c |
| Find Angle A | A = acos((b² + c² – a²) / 2bc) | Sides a, b, c |
Cosine Rule FAQ
Learn more about the Law of Cosines, triangle solving, and trigonometric applications.
The Cosine Rule (or Law of Cosines) is a fundamental formula in trigonometry that relates the lengths of the three sides of a triangle to the cosine of one of its angles. It is used to find missing sides or angles in any triangle, not just right-angled ones.
You should use the Law of Cosines when you know either two sides and the included angle (SAS) to find the third side, or when you know all three sides (SSS) to find any of the angles. It generalizes the Pythagorean theorem for non-right triangles.
To find a side, the formula is c² = a² + b² – 2ab cos(C). To find an angle, the formula is rearranged to cos(C) = (a² + b² – c²) / 2ab, where C is the angle opposite side c.
Yes, but it simplifies to the Pythagorean theorem (a² + b² = c²). Since the cosine of 90 degrees is 0, the term ‘2ab cos(C)’ becomes zero, leaving c² = a² + b². For right triangles, the Pythagorean theorem or basic trigonometry (SOH CAH TOA) is usually faster.
In the standard formula c² = a² + b² – 2ab cos(C), the angle C is always the angle opposite to the side c. Similarly, angle A is opposite side a, and angle B is opposite side b. The included angle between sides a and b is always angle C.