Find a missing triangle side from two sides and the included angle
Result
Missing Side
Step-by-Step Derivation
Law of Cosines Formula
c² = a² + b² - 2ab cos(C)
The law of cosines extends the Pythagorean theorem to any triangle. It uses two known sides and the included angle to calculate the opposite side, so it is useful when a triangle is not right-angled.
⚠Enter the included angle in degrees. The angle must be greater than 0 and less than 180, and both side lengths must be positive.
How the Law of Cosines Works
The cosine term adjusts the side length based on the angle between the known sides. When the angle is 90 degrees, cos(C) is zero and the formula becomes the Pythagorean theorem.
Included Angle
C is the angle between sides a and b.
Opposite Side
The result c is across from angle C.
Right Triangle Case
At 90 degrees, the cosine term disappears.
Triangle Solving
Useful for SAS and SSS triangle problems.
💡 Example: For a=7, b=9, and C=60°, c²=49+81-126cos(60°)=67, so c=8.185.
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