Formula
s = (a + b + c) ÷ 2Area = √(s(s − a)(s − b)(s − c)) (Heron’s formula)cos A = (b² + c² − a²) ÷ 2bc (law of cosines) How it works
Three side lengths fix a triangle completely, so you can work out everything else about it. Heron’s formula gives the area directly from the sides, with no need to find a height first: compute the semi-perimeter s, multiply s by its differences from each side, and take the square root.
The angles come from the law of cosines, a generalisation of the Pythagorean theorem that works for any triangle. Once two angles are known, the third is whatever is left of 180°.
Not every set of three lengths makes a triangle. The triangle inequality says each side must be shorter than the sum of the other two; 2, 3 and 6 fail because 2 + 3 < 6, so the short sides can’t meet.
Frequently asked questions
How can I tell if a triangle is right-angled from its sides?
Check whether the squares of the two shorter sides add up to the square of the longest. If a² + b² = c², the angle opposite c is exactly 90°. If the sum is smaller, the triangle is obtuse; if larger, acute.
What if I know two sides and an angle instead?
Use the law of cosines to find the third side first (c² = a² + b² − 2ab·cos C), then enter all three sides here.