How this calculator works
The cosine term adjusts the Pythagorean relationship for the angle between two sides. It vanishes at a right angle, is positive for acute angles and is negative for obtuse angles. Three known sides uniquely determine the angles if the strict triangle inequality holds. Calculations retain full internal precision; choose degrees, radians or turns in the readout.
Formula
c²=a²+b²-2ab cos C; cos A=(b²+c²-a²)/(2bc); cos B=(a²+c²-b²)/(2ac); K=ab sin C/2.
Worked example
Sides a = 3 m, b = 4 m and C = 90° give c = 5 m and area 6 m². Equal sides a = b = 2 cm with C = 60° give c = 2 cm and area √3 cm².
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Frequently asked questions
Why must C be the included angle?
The formula relates c to the angle between a and b. If the angle is opposite one of the known sides, use the law of sines and check for ambiguity.
What if the three sides cannot meet?
The calculator rejects zero lengths and lengths whose two shortest sides sum to no more than the longest.
Does the law work for an obtuse triangle?
Yes. The cosine becomes negative, so the opposite side squared exceeds the sum of the other two squares.