Law of Cosines Calculator
How It Works
This law of cosines calculator uses established formulas to provide accurate results.
The basic rule:
- c² = a² + b² - 2ab·cos(C)
- cos(A) = (b² + c² - a²) / (2bc)
- cos(B) = (a² + c² - b²) / (2ac)
- Area = ½ab·sin(C)
Results are estimates. Consult a professional for critical decisions.
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Explore Allday Everyday Math →Frequently Asked Questions
What is the Law of Cosines?
The Law of Cosines states c² = a² + b² - 2ab·cos(C). It generalizes the Pythagorean theorem to all triangles. When C = 90°, cos(90°) = 0 and it reduces to c² = a² + b².
When should I use the Law of Cosines?
Use it when you know: (1) Two sides and the included angle (SAS) to find the third side, or (2) All three sides (SSS) to find an angle. Rearrange to find angles: cos(C) = (a² + b² - c²)/(2ab).
How is the Law of Cosines related to the Pythagorean theorem?
The Pythagorean theorem is a special case of the Law of Cosines. When angle C is exactly 90°, cos(C) = 0, so the -2ab·cos(C) term disappears, leaving c² = a² + b².
Can I find all three angles from three sides?
Yes. Use the Law of Cosines three times to find each angle: cos(A) = (b²+c²-a²)/(2bc), cos(B) = (a²+c²-b²)/(2ac), and cos(C) = (a²+b²-c²)/(2ab). Or find two angles and use A+B+C=180° for the third.