Law of Cosines Calculator

How to Use This Calculator

1. For SAS: enter two sides and the included angle to find the missing side.
2. For SSS: enter all three sides to find all angles.
3. Use the Generalized tab to explore the Pythagorean and vector connections.

Formula

Example

Frequently Asked Questions

• What is the Law of Cosines? ▼
  The Law of Cosines states c² = a² + b² − 2ab·cos(C). It relates all three sides of a triangle to one of its angles, and generalises the Pythagorean theorem to any triangle.
• When should I use the Law of Cosines? ▼
  Use it when you know SAS (two sides and their included angle) to find the third side, or SSS (all three sides) to find any angle. For AAS or ASA use the Law of Sines instead.
• How is it related to the Pythagorean theorem? ▼
  When C = 90°, cos(C) = 0, so c² = a² + b². The Law of Cosines reduces to the Pythagorean theorem as a special case.
• How do I find angles using the Law of Cosines? ▼
  Rearrange: cos(A) = (b² + c² − a²) / (2bc). Then A = arccos of that value. Use this for SSS to find all three angles.
• What is Heron's formula and how is it related? ▼
  Heron's formula computes area from three sides: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2. It gives the same area as ½ab·sin(C) computed from the Law of Cosines approach.

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