Scalene Triangle

A scalene triangle is a triangle where all three sides have different lengths and all three angles are different, making it the most general type of triangle. Unlike isosceles or equilateral triangles, scalene triangles have no symmetry -- each side and angle is unique. This calculator solves any scalene triangle from just the three side lengths using Heron's formula for area and the Law of Cosines for angles, making it ideal for students studying geometry, engineers working with irregular triangular structures, architects designing non-standard shapes, and anyone who needs to compute triangle properties from raw measurements.

The area is computed via Heron's formula: A = sqrt(s(s-a)(s-b)(s-c)), where a, b, c are the three side lengths and s = (a + b + c)/2 is the semi-perimeter. For example, a triangle with sides 5, 6, and 7 units has a semi-perimeter s = 9, giving an area of sqrt(9 x 4 x 3 x 2) = sqrt(216) ≈ 14.70 square units. The angles are found using the Law of Cosines: angle A = arccos((b^2 + c^2 - a^2) / (2bc)), angle B = arccos((a^2 + c^2 - b^2) / (2ac)), and angle C = 180 deg - angle A - angle B. For the same 5-6-7 triangle, the angles are approximately 44.4 deg, 57.1 deg, and 78.5 deg, which always sum to 180 deg. The perimeter is simply a + b + c -- for the example triangle, that is 18 units.

The calculator first validates your inputs using the triangle inequality theorem, which states that the sum of any two sides must exceed the third side. If this condition is not met (e.g., sides 1, 2, 3), the lengths cannot form a triangle and the calculator returns an error. For a valid scalene triangle, all three sides must also be different lengths. If two or three sides are equal, the triangle is isosceles or equilateral, not scalene.

Regional Notes: Triangle geometry is universal and independent of any regional standards. You may use any unit system -- metric (cm, m, km) or imperial (in, ft, mi) -- as long as all three side lengths use the same unit. The area is reported in square units of the input unit, and the perimeter is in the same linear units. Angles are always reported in degrees, which are the standard unit for angle measurement worldwide. The triangle inequality theorem (a + b > c, a + c > b, b + c > a) applies universally regardless of region or unit system.