Height

Definition

Triangles

The height of a triangle is the perpendicular line from one vertex to its opposite side, which is arbitrarily denoted as the base. For example, in the following diagram, the height is the highlighted portion of the triangle below. $[asy] size(5cm); dot((0,0)); dot((5,0)); dot((3,4)); draw((0,0)--(5,0)--(3,4)--cycle); draw((3,0)--(3,4),red); [/asy]$

The height is of importance in triangle geometry, because we have $A=\frac{1}{2}bh$, where A is the area, b is the length of the base, and h is the length of the height.

General

The height is the measure of how high something is. In three-dimensional geometry, it is the distance from the "highest" point to the "lowest" point along the direction which we use to define "highest" and "lowest". For example, the height of a pyramid is the distance from the central vertex to the opposite base.

The height is still of importance in three-dimensional geometry because it can be used to calculate volume. For a prism, we have $V=Bh$, where V is the volume, B is the base's area, and h is the height's length; for a pyramid, we have $V=\frac{1}{3}Bh$.