Free · Geometry and function relationships

Triangle Angles from Sides or Included Angles

Use the law of cosines with three sides or two sides and their included angle. Interior angles must be positive and sum to 180°; each side must be shorter than the other two combined.

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

For given = 0, sideLength_a = 3, sideLength_b = 4, sideLength_c = 5, angle_alpha = 36.86989764584402129685561255909341065759157140070955794402796738, angle_beta = 53.13010235415597870314438744090658934240842859929044205597203262, angle_gamma = 90.

cos(α)=(b²+c²−a²)/(2bc)

Worked example

Enter these known values and leave the other values blank.

Given
0
Side length a
3 cm
Side length b
4 cm
Side length c
5 cm
Angle α
36.87 deg
Angle β
53.13 deg
Angle γ
90 deg

Assumptions and limitations

  • Use the law of cosines with three sides or two sides and their included angle. Interior angles must be positive and sum to 180°; each side must be shorter than the other two combined.
  • Clear the unknown field before solving. Locks preserve known quantities; an inverse may have multiple solutions and this model selects its principal branch.
  • Numerical rounding does not describe measurement uncertainty. Extremely large arguments or near-singular inverses can exceed the available precision; use moderate values for interpretable finite results.

Common questions

How do I enter a consistent set of values?

Use the law of cosines with three sides or two sides and their included angle. Interior angles must be positive and sum to 180°; each side must be shorter than the other two combined.

Can I calculate a missing input?

Clear that field and provide enough independent known quantities. Output-only fields and nonunique inverse branches may need a different mode.

References

A nondegenerate triangle requires all three strict triangle inequalities. The captured source omits one or more explicit checks, allowing impossible SSS inputs to leave calculated angles blank without an error.

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