Mathematics · Trigonometry
Law of Cosines Side Calculator
Find a triangle side from two sides and their included angle.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- c=√(3²+4²−2×3×4cos90°)=5.
Problem → model → reason → result
What problem does this model solve?
Find a triangle side from two sides and their included angle.
Why does the model apply?
The cosine law generalizes the Pythagorean theorem to non-right triangles.
What assumptions does it make?
The angle convention shown by the inputs is used consistently and the triangle or circular relationship matches the stated model.
Formula
c=√(a²+b²−2ab cos C)
Calculation and working
The calculator above substitutes your inputs into the model and leaves the calculation steps visible so the answer can be checked rather than merely accepted.
What does the result mean?
The result answers the stated problem in the units implied by your inputs. Read its sign, size and units together, then compare it with the original values before drawing a conclusion.
Worked example
Sides 3 and 4 with 90° between them give side 5.
Common mistake
Use the angle between the two entered sides.
When does this model not apply?
Ambiguous triangle data, rounded angles and the wrong degree/radian mode can produce a plausible-looking but incorrect result.
How to learn with this calculator
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Clear answers
Frequently asked questions
What does the Cosine law side do?
Find a triangle side from two sides and their included angle.
How does the Cosine law side work?
The calculator applies c=√(a²+b²−2ab cos C). The cosine law generalizes the Pythagorean theorem to non-right triangles.
What can I learn from the Cosine law side?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed 2026-07-14. Calculations tested 2026-07-14.