# Altitude

The altitude defined is as in a mathematical sense, there are more definitions of the word altitude not mentioned here

## Definition

In geometry, an altitude of a figure is a cevian that is perpendicular to the side to which it extends.

Usually, one is concerned with the altitude (or height) of triangles. In particular, the altitudes of any triangle are concurrent at a point known as the orthocenter.

## Usage

Altitudes are mainly used to finding the area of a triangle, frequently used as the height in $\frac{1}{2} bh$. Furthermore, if one knows the area of a triangle and one side, the altitude to that side can be calculated. In addition to triangles, altitudes are also useful for finding the area of quadrilaterals with at least one pair of parallel sides (rectangles, parallelograms, trapezoids).

Because the altitude is always perpendicular to a side, creating right angles, many right triangle tools like the Pythagorean Theorem can be used and can be very useful.

## Diagram $[asy] size(15cm); markscalefactor = 0.01; dot((0,0)); dot((4,0)); dot((3,3)); dot((3,0)); dot((2,2)); dot((3,1)); dot((3.6,1.2)); draw((0,0)--(4,0)--(3,3)--cycle); draw((0,0)--(3.6,1.2),red); draw((2,2)--(4,0),red); draw((3,0)--(3,3), red); draw(rightanglemark((0,0),(3,0),(3,1))); draw(rightanglemark((4,0),(3.6,1.2),(0,0))); draw(rightanglemark((0,0),(2,2),(4,0))); label("Orthocenter",(2.85,1.1),W); [/asy]$ The lines highlighted are the altitudes of the triangle, they meet at the orthocenter, typically labeled $H.$