Scalene Triangle Calculator

A scalene triangle features three sides of completely different lengths (a, b, and c) and three unique interior angles. When you know all three side lengths, the area is found using Heron's formula: Area = 0.25 * sqrt( (a+b+c) * (-a+b+c) * (a-b+c) * (a+b-c) ). The interior angles are determined via the Law of Cosines, such that cos(angle_ab) = (a^2 + b^2 - c^2) / (2 * a * b). For right-angled scalene triangles, simpler rules apply using the legs and the hypotenuse, alongside standard height formulas where height equals twice the area divided by the respective base length.

Let's calculate the perimeter, area, and angles of a standard scalene triangle with side lengths a = 7 cm, b = 10 cm, and c = 12 cm. First, find the perimeter by summing all three sides: Perimeter = 7 + 10 + 12 = 29 cm. Next, apply Heron's formula for the area: semi-perimeter s = 29 / 2 = 14.5 cm. Area = sqrt(14.5 * (14.5 - 7) * (14.5 - 10) * (14.5 - 12)) = sqrt(14.5 * 7.5 * 4.5 * 2.5) = sqrt(1223.4375) approx 34.98 square centimeters. Finally, use the Law of Cosines to find the angle opposite to side c: cos(angle) = (7^2 + 10^2 - 12^2) / (2 * 7 * 10) = (49 + 100 - 144) / 140 = 5 / 14 approx 0.3571, meaning the angle is approximately 69.08 degrees.

Always verify that your input side lengths satisfy the triangle inequality theorem (the sum of any two sides must be strictly greater than the third side) before attempting calculations. Pay close attention to unit consistency; mixing centimeters and meters will result in massive calculation errors. When dealing with right scalene triangles, always correctly identify the hypotenuse as the side opposite the ninety-degree angle to avoid misapplying trigonometric functions.