Formula and Method for Classifying an Obtuse Triangle
A triangle is classified by its largest interior angle: acute if that angle is less than 90°, right if it equals 90°, and obtuse if it is greater than 90°. Given only the three side lengths (a, b, c), this calculator uses the Law of Cosines to solve for every interior angle, determines the classification, and also reports the triangle's area and perimeter.
How the calculation works
The Law of Cosines generalizes the Pythagorean theorem to any triangle: c² = a² + b² − 2ab·cos(C), where C is the angle opposite side c. Rearranged to solve for the angle, this becomes cos(C) = (a² + b² − c²) / (2ab). The calculator applies this formula to find two of the three angles directly, then finds the third by subtracting the other two from 180° (since a triangle's interior angles always sum to 180°). Whichever angle turns out largest determines the classification — and because the longest side is always opposite the largest angle, that largest angle sits opposite whichever of a, b, or c you entered as the longest side. Area is found separately with Heron's formula: Area = √[s(s−a)(s−b)(s−c)], where s = (a + b + c) / 2 is the semi-perimeter — this works for every triangle regardless of its angles.
Common mistakes
- Invalid side lengths: not every three numbers form a triangle. Each side must be shorter than the sum of the other two (the triangle inequality) — for example, 2, 3, and 10 cannot form a triangle.
- Mixing units: enter all three sides in the same unit; mixing inches with feet produces a meaningless answer.
- Assuming two obtuse angles are possible: they are not — since all three angles sum to 180°, only one angle can ever exceed 90°.
- Rounding near 90°: if the largest angle is extremely close to 90° (a near-right triangle), small measurement errors in the side lengths can flip the classification between acute, right, and obtuse.
Real-world applications
- Structural engineering uses angle classification to evaluate roof trusses, braces, and load paths, since obtuse joints behave differently under load than acute or right ones.
- Surveying and navigation use the Law of Cosines to compute unknown angles or distances from measured baselines (triangulation).
- Carpentry and design use triangle classification to confirm a cut or layout matches an intended shape before materials are committed.
- Physics and computer graphics use Heron's formula and the Law of Cosines constantly for collision geometry, mesh analysis, and force diagrams.