How to Find the Height of a Triangle
The height (or altitude) of a triangle is the perpendicular distance from a vertex to the opposite side (the base). Every triangle has three altitudes, one from each vertex. The height is essential for computing area and solving many geometry problems.
Height Formulas
From Area and Base
Rearrange A = (1/2) x b x h to solve for h.
h = 2A / b
From Three Sides (Heron's)
First find the area via Heron's formula, then derive each height.
h_a = 2A / a, h_b = 2A / b, h_c = 2A / c
Heron's Formula
Compute area from three sides: s = (a+b+c)/2
A = sqrt(s(s-a)(s-b)(s-c))
Why Heights Matter
- Heights are used to calculate the area of a triangle.
- The orthocenter is the point where all three altitudes intersect.
- In right triangles, the altitude to the hypotenuse creates similar triangles.
- Heights are critical in structural engineering and architecture for load calculations.