How to Calculate Triangle Area and Perimeter
This calculator determines the area (using Heron's formula), perimeter, and type by sides and by angles of a triangle, given three sides.
Heron's Formula
Heron's formula calculates the area of a triangle knowing only the three side lengths, without needing to know heights or angles:
A = sqrt(s x (s-a) x (s-b) x (s-c))
where s = (a + b + c) / 2 is the semi-perimeter. This formula was published by Heron of Alexandria in the 1st century AD in his work Metrica, but was probably already known to Archimedes.
Triangle Classification
By sides: Equilateral (three equal sides, three 60-degree angles), Isosceles (two equal sides, two equal base angles), Scalene (all sides different, all angles different).
By angles: Right (one 90-degree angle -- Pythagorean theorem applies), Acute (all angles less than 90 degrees), Obtuse (one angle greater than 90 degrees).
The calculator determines the type using the law of cosines: if a² + b² = c² (where c is the longest side) the triangle is right; if a² + b² > c² it is acute; if a² + b² < c² it is obtuse.
The Triangle Inequality
Not all three positive numbers can form a triangle. The triangle inequality must hold: the sum of any two sides must be strictly greater than the third side.
The Triangle in Geometry
The triangle is the simplest polygon and the fundamental building block of geometry. Every polygon can be decomposed into triangles (triangulation). This property is the basis of 3D computer graphics, where every surface is approximated with triangles.