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Law of Cosines Calculator.

Solve the side opposite angle A from two sides and their included angle.

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Set your values

Results update as you type.

Solved side a: 6.244998 units

Solved side a

0.000000units
Angle A: 60

Angle A

0.000000

Results update automatically as you type.

Use Cases

Find the third side of a triangle

When you know two sides and the included angle (SAS), use this calculator to find the length of the third side without needing to measure it directly.

Example: If b=5, c=7, and A=60°, the calculator gives a≈6.24.

Check triangle geometry problems

Verify your manual calculations for homework or design projects that involve triangles, ensuring accuracy in side lengths.

Example: Double-check a triangle with b=8, c=10, A=45°.

Frequently Asked Questions

What is the Law of Cosines?
The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides b, c, and a, and angle A opposite side a, it states: a² = b² + c² - 2bc·cos(A).
How do I use this calculator?
Enter the lengths of side b and side c, and the measure of the included angle A in degrees. The calculator will compute the length of side a using the Law of Cosines formula.
What if my angle is in radians?
This calculator expects the angle A in degrees. If you have the angle in radians, convert it to degrees by multiplying by 180/π before entering it.

Tips & Common Mistakes

Tips

  • Ensure that the angle A is the angle between side b and side c, as the formula uses the included angle.
  • Use consistent units for side b and side c; the result will be in the same unit.
  • For angles greater than 90°, the cosine is negative, which affects the calculation correctly.
  • If you need to find an angle instead of a side, rearrange the formula or use a different calculator.

Common Mistakes to Avoid

  • Entering the angle in radians instead of degrees, leading to incorrect results.
  • Using the wrong angle – the included angle must be between the two given sides.
  • Forgetting to square the side lengths when applying the formula manually.

Last updated: August 13, 2026