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Triangle Sides and Angles with Cosine-Law Root Selection

No. An angle, its opposite side and another side can describe two triangles. This workflow uses the captured plus-square-root side branch and rejects nonreal or nonpositive branches. It does not enumerate every alternative.

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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.

Private by default

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

Formula and method

Example in initial displayed units: Side (a) 3, Side (b) 4, Side (c) 5 gives Angle α 36.8699, Angle β 53.1301, Angle γ 90.

a² = b²+c²−2bc cos α; α+β+γ = π

Worked example

Enter these known values and leave the other values blank.

Side (a)
3 mm
Side (b)
4 mm
Side (c)
5 mm
Angle α
36.9 deg
Angle β
53.1 deg
Angle γ
90 deg

Assumptions and limitations

  • Positive sides form a nondegenerate Euclidean triangle, with α opposite a, β opposite b and γ opposite c. Retained variants support SSS, SAS and selected SSA configurations. An SSA side inverse chooses the plus-square-root branch; another positive triangle may also fit the same measurements. Two angles and one side do not recover all sides with these captured dependency rules, even where the geometry is uniquely determined. Three angles do not fix scale.
  • Clear unknowns before solving backward and select compatible units. Original reference drawings do not scale to entered values. Independently rounded boundary measurements can be incompatible at finite precision.
  • The source strict-inequality checks use finite numerical comparison; extremely nearly degenerate measurements can be rejected. The ordered half-angle evaluation improves slender-triangle angles without changing the positive-root side choice.

Common questions

Does an SSA entry always define one triangle?

No. An angle, its opposite side and another side can describe two triangles. This workflow uses the captured plus-square-root side branch and rejects nonreal or nonpositive branches. It does not enumerate every alternative.

How many measurements are needed?

Use the worked example or a verified independent input pair or triple. Supplying dependent measurements does not add enough information; a consistent shape still needs a length scale.

References

Require real cosine-law SSA side roots; a positive discriminant does not remove the documented two-triangle ambiguity. Evaluate cosine-law angles with the equivalent ordered half-angle atan2 form, avoiding loss of precision in acos of a value very close to one for needle-like triangles.

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