Side a

in

Side b

in

Side c

in

Angle α

deg

Angle β

deg

Angle γ

deg

The law of cosines calculator can help you to solve problems that include triangles. You will learn the law of cosines formula and its applications. Thanks to this triangle calculator, you will be able to quickly find the properties of any arbitrary triangle.

The law of cosines describes the relationship between the lengths of the sides of any triangle. This law generalizes the Pythagorean theorem, as it allows to calculate the length of one of the sides, knowing the other sides. It can be applied to all triangles, not only the right triangles.

The law of cosines states that, for a triangle with sides and angles denoted with symbols as in the picture above,

`a² = b² + c² - 2bc * cos(α)`

`b² = a² + c² - 2ac * cos(β)`

`c² = a² + b² - 2ab * cos(γ)`

You can transform these law of cosines formulas to solve some problems of triangulation (solving a triangle).You can use them to find:

- The third side of a triangle, knowing two sides and the angle between them:

`a = √[b² + c² - 2bc * cos(α)]`

`b² = √[a² + c² - 2ac * cos(β)]`

`c² = √[a² + b² - 2ab * cos(γ)]`

- The angles of a triangle, knowing all three sides:

`α = arccos [(b² + c² - a²)/(2bc)]`

`β = arccos [(a² + c² - b²)/(2ac)]`

`γ = arccos [(a² + b² - c²)/(2ab)]`

- The third side of a triangle, knowing two sides and an angle opposite to one of them:

`a = b*cos(γ) ±√[c² - b²*sin²(γ)]`

`b = c*cos(α) ±√[a² - c²*sin²(α)]`

`c = a*cos(β) ±√[b² - a²*sin²(β)]`

You can combine the law of cosines calculator with the law of sines to solve other problems, for example finding the sides of the triangle, knowing all of the angles.

- Start with formulating your problem. For example, you may know all sides of the triangle and look for the angles.
- Input the known values into the appropriate boxes of this triangle calculator. Remember to double-check with the figure above whether you denoted the sides and angles with correct symbols.
- Watch our law of cosines calculator performing all calculations for you!

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