Triangle Height Calculator

The height (or altitude) of a triangle is a line segment drawn from a vertex that is perpendicular to the opposite side, known as the base. The primary formula derives from the classic area of a triangle equation: Area = (1/2) * base * height. By rearranging this formula, the height relative to a specific side can be found as h = (2 * Area) / base. For specialized triangles, different formulas apply. For instance, in an equilateral triangle with side length a, the height is calculated using the Pythagorean theorem, simplifying to h = (a * sqrt(3)) / 2. For right triangles, the legs themselves can serve as heights depending on which side is chosen as the base.

Let us find the height of a scalene triangle with side lengths a = 7 cm, b = 9 cm, and c = 12 cm. First, we determine the semi-perimeter s = (7 + 9 + 12) / 2 = 14 cm. Using Heron's formula to find the area: Area = sqrt(14 * (14 - 7) * (14 - 9) * (14 - 12)) = sqrt(14 * 7 * 5 * 2) = sqrt(980) approximately equal to 31.304 sq cm. Next, to find the height relative to side c (base = 12 cm), we apply the altitude formula: h = (2 * 31.304) / 12 = 62.608 / 12 = 5.22 cm. Thus, the perpendicular height from the base of 12 cm to the opposite vertex is 5.22 centimeters.

Always ensure your input measurements use the same unit of length (such as centimeters or inches) before performing calculations. When dealing with right triangles, clearly identify which sides form the 90-degree angle, as they represent the base and perpendicular height when no other altitude is given. Double-check angle inputs to confirm whether your calculator expects degrees or radians to prevent unexpected computational results.