Triangle Solver

At the heart of this calculator is the Law of Cosines, which generalizes the Pythagorean theorem to any triangle regardless of its shape. Given sides a, b, and c, each angle is derived using the formula: cos(A) = (b² + c² − a²) / (2bc), and similarly for angles B and C. Once all three angles are found, the calculator verifies they sum to exactly 180°, catching any rounding or input errors. This approach works universally — whether your triangle is acute, right, or obtuse — whereas simpler methods like basic trigonometry ratios only apply in specific cases.

For the area, the calculator applies Heron's Formula, which is ideal because it requires only the three side lengths with no need to measure height. First, the semi-perimeter s = (a + b + c) / 2 is computed. Then the area is calculated as √(s(s−a)(s−b)(s−c)). Heron's Formula is remarkably elegant and numerically stable for most practical inputs, making it the preferred method when only side lengths are known — a common real-world scenario in surveying, land measurement, and construction.

The triangle's type is determined automatically from the computed angles and side relationships. If all three angles are less than 90°, it's acute. If one angle equals exactly 90° (within a small tolerance for floating-point arithmetic), it's a right triangle. If one angle exceeds 90°, it's obtuse. Side-length comparisons simultaneously classify the shape as equilateral (all sides equal), isosceles (exactly two sides equal), or scalene (all sides different). This dual classification gives users a complete geometric picture without any manual inspection.

A critical thing to understand is the triangle inequality: for any valid triangle, the sum of any two sides must be strictly greater than the third side. If you enter values like 1, 2, and 10, no triangle can exist — and the calculator will flag this immediately rather than returning nonsensical results. A common mistake users make is entering dimensions from different units (mixing centimeters and inches, for example), which violates the implicit assumption that all three sides share the same unit. Always ensure consistent units before solving, and remember that the area output will be in square units of whatever length unit you input.