Math

Law of Cosines Calculator

Enter two sides and the included angle (SAS) to solve the rest of the triangle.

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Your numbers

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Side c
Side c
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Angle A (°)
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Angle B (°)
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Area
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Law of Cosines Calculator by TheCalcTools

How the Law of Cosines Calculator works

Enter two sides and the angle between them.

The law of cosines gives the third side: c² = a² + b² − 2ab cos C.

The other two angles then follow from the same law.

The formula

c² = a² + b² − 2ab·cos CArea = ½ ab sin C

Worked example

Using the default values shown in the calculator:

Side a7
Side b10
Included angle C (°)40
  1. c² = 49 + 100 − 2 × 7 × 10 × cos 40° = 149 − 107.246 = 41.754.
  2. c = 6.4617.
  3. Angle A = 44.13° and angle B = 95.87°.

Result:

Side c6.4617
Angle A (°)44.13
Angle B (°)95.87
Area22.4976

Frequently asked questions

How is it related to the Pythagorean theorem?

When C = 90° the cosine term is zero and the law of cosines reduces to a² + b² = c².

Can it find an angle from three sides?

Yes. Rearrange to cos C = (a² + b² − c²) ÷ 2ab. The triangle area calculator does this for you.

Sources and further reading

Triangle Area Calculator (Heron's Formula)Enter three side lengths to find the triangle's area, perimeter and angles Pythagorean Theorem CalculatorSolve for any side of a right triangle using a² + b² = c² Degrees to Radians ConverterConvert an angle between degrees and radians and see its sine, cosine and tangent Trigonometry Calculator (sin, cos, tan, sec, csc, cot)Enter an angle and choose degrees or radians

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