Law of Cosines Calculator
Calculate triangle sides, angles, area, and perimeter using the Law of Cosines formula
Two Sides and Included Angle (SAS)
Enter two sides and the angle between them to calculate the remaining side and angles.
Calculation Results
Side Length Proportions
Triangle Diagram
Calculation Steps
Enter values and click Calculate to see step-by-step calculations.
Three Sides (SSS)
Enter all three sides to calculate the angles, perimeter, and area.
Calculation Results
Two Sides and Non-Included Angle (SSA)
Enter two sides and an angle not between them. Note: This case may have 0, 1, or 2 possible solutions (ambiguous case).
Calculation Results
Enter values and click Calculate to see possible solutions.
Law of Cosines Formula
The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles:
c² = a² + b² – 2ab cos(C)
Where:
- a, b, c are the sides of the triangle
- C is the angle opposite side c
This formula is useful for:
- Finding the third side when two sides and the included angle are known (SAS)
- Finding the angles when all three sides are known (SSS)
- Solving triangles in cases where the Law of Sines cannot be applied
Example Calculations
Example 1: SAS Case
Given: a = 5, b = 7, C = 60°
Find side c:
c² = 5² + 7² – 2×5×7×cos(60°)
c² = 25 + 49 – 70×0.5
c² = 74 – 35
c² = 39
c = √39 ≈ 6.245
Example 2: SSS Case
Given: a = 3, b = 4, c = 5
Find angle C:
cos(C) = (a² + b² – c²) / (2ab)
cos(C) = (9 + 16 – 25) / (2×3×4)
cos(C) = 0 / 24
cos(C) = 0
C = cos⁻¹(0) = 90°