△ Triangle Calculator

Triangle Calculations: Sides, Angles, and Area

Every triangle has three sides and three angles that always sum to 180°. Given enough information, you can calculate everything else. The minimum input: three pieces of data, at least one of which must be a side length. The three key tools are the Pythagorean theorem (right triangles only), the Law of Sines (any triangle: a/sin A = b/sin B = c/sin C), and the Law of Cosines (any triangle: c² = a² + b² − 2ab·cos C).

Right Triangles and the Pythagorean Theorem

In a right triangle, the hypotenuse squared equals the sum of the squares of the other two sides: a² + b² = c². The hypotenuse is always the longest side, opposite the 90° angle. The most famous Pythagorean triples (whole-number solutions) are 3-4-5, 5-12-13, and 8-15-17. Trigonometric ratios are defined for right triangles: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent. These ratios let you find unknown sides from known angles, or unknown angles from known sides.

Area When No Height Is Given: Heron's Formula

The standard area formula ½ × base × height requires knowing a height, which isn't always given. Heron's formula calculates area from just the three side lengths: compute the semi-perimeter s = (a+b+c)/2, then area = √(s(s−a)(s−b)(s−c)). This works for any triangle — right, acute, or obtuse. Named after Hero of Alexandria (around 60 AD), it remains one of the most elegant results in classical geometry.