Free · Mathematical relations and geometry
Triangle Side and Angles by the Cosine Law
Find a triangle’s third side from two sides and their included angle, or determine all angles from three sides.
- Formula & worked example
- Private in your browser
- No signup
Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- 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
Sides 3 and 4 enclosing a 90° angle give an opposite side of 5. In reverse SSS mode, a 3–4–5 triangle has a right angle opposite side 5.
c² = a² + b² − 2ab cos γ
Worked example
Enter these known values and leave the other values blank.
- Given
- 0
- Side a
- 3 cm
- Side b
- 4 cm
- Angle γ
- 90 deg
- Side c
- 5 cm
Assumptions and limitations
- Sides are positive and interior angles lie strictly between zero and 180 degrees.
- SSS requires strict triangle inequalities. SAS uses the included angle between the two known sides.
- The calculator provides forward SAS side and forward SSS angles; outputs are read-only.
Common questions
How can I solve for a missing quantity?
Leave the unknown control empty and enter the quantities needed by the selected relation. Read the assumptions for any branch or domain restrictions.
How should I interpret the worked example?
It is an exact mathematical model with independently checked numbers. Display rounding does not change the values used for subsequent calculations.
References
The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.
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