Triangle Calculator

The Triangle Calculator is a comprehensive geometric tool that computes all essential properties of any triangle when you know the lengths of its three sides. Whether you are a student solving geometry homework, an engineer designing structural components, or an architect planning floor layouts, this calculator delivers instant and accurate results for area, perimeter, all interior angles, and the classification of your triangle.

Triangles are the most fundamental polygon in Euclidean geometry. Every polygon with more than three sides can be decomposed into triangles, making the triangle the building block of all planar geometry. The study of triangles dates back to ancient civilizations: Babylonian clay tablets from around 1800 BCE contain problems about triangles, and the Egyptians used the 3-4-5 right triangle to survey land after the annual Nile floods. Greek mathematicians, particularly Euclid and Heron of Alexandria, formalized the relationships between sides and angles that we still use today.

This calculator uses Heron's formula to determine the area from three sides alone, without requiring knowledge of any angle or height. Heron's formula, attributed to Heron of Alexandria (c. 10-70 CE), is elegant in its simplicity: first compute the semi-perimeter, then apply a single square-root expression. The formula works for every valid triangle, regardless of its shape or orientation.

For the interior angles, the calculator employs the Law of Cosines, a generalization of the Pythagorean theorem that applies to all triangles. Given three side lengths, the Law of Cosines yields the cosine of each angle, from which the angle itself is recovered using the inverse cosine function. The three computed angles always sum to exactly 180 degrees, providing a useful consistency check.

Beyond raw measurements, the calculator classifies your triangle by two criteria simultaneously. By side lengths, a triangle is equilateral (all sides equal), isosceles (exactly two sides equal), or scalene (all sides different). By angles, a triangle is right (one 90-degree angle), obtuse (one angle greater than 90 degrees), or acute (all angles less than 90 degrees). These combined classifications yield descriptors such as "Isosceles Right" or "Scalene Obtuse" that fully characterize the triangle's geometry.

Practical applications of triangle calculations span virtually every technical field. In civil engineering, triangulated trusses distribute loads efficiently because triangles are inherently rigid. In surveying and navigation, triangulation determines unknown positions from known reference points. In computer graphics, meshes of triangles approximate curved surfaces for rendering. In physics, vector decomposition relies on right-triangle relationships. Even in everyday life, tasks like cutting fabric, tiling floors, or estimating the area of an irregularly shaped garden plot often reduce to triangle calculations.

Enter the three side lengths below, and the calculator will instantly provide the area, perimeter, all three interior angles, and the triangle's classification. For valid results, ensure that the side lengths satisfy the triangle inequality: the sum of any two sides must exceed the third side.