Free · Geometry and function relationships

Triangle Side and Angle Relationships

Match each angle to the side opposite it. The law of cosines calculates the missing side from two known sides and their included angle; the angle-sum mode finds the third interior angle.

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Calculator inputs

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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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Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

For given = 0, side_a = 3, side_b = 4, side_c = 5, angle_alpha_0 = 36.86989764584402129685561255909341065759157140070955794402796738, angle_beta_0 = 53.13010235415597870314438744090658934240842859929044205597203262, angle_gamma_0 = 90.

a²=b²+c²−2bc cos(α)

Worked example

Enter these known values and leave the other values blank.

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

Assumptions and limitations

  • Match each angle to the side opposite it. The law of cosines calculates the missing side from two known sides and their included angle; the angle-sum mode finds the third interior angle.
  • 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?

Match each angle to the side opposite it. The law of cosines calculates the missing side from two known sides and their included angle; the angle-sum mode finds the third interior angle.

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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