# Altitude of a Triangle

The **altitude of a triangle**, or **height**, is a line from a vertex to the opposite side, that is perpendicular to that side. It can also be understood as the distance from one side to the opposite vertex.

Every triangle has three altitudes (h_{a}, h_{b} and h_{c}), each one associated with one of its three sides. If we know the three sides (*a*, *b*, and *c*) it’s easy to find the three altitudes, using the Heron’s formula:

## Altitudes and Orthocenter

The three **altitudes** of a triangle (or its extensions) intersect at a point called orthocenter.

The altitude can be inside the triangle, outside it, or even coincide with one of its sides, it depends on the type of triangle it is:

- Obtuse triangle: The altitude related to the longest side is inside the triangle (see h
_{c}, in the triangle above) the other two heights are outside the triangle (h_{a}, and h_{b}). - Right triangle: The altitude with respect to the hypotenuse is interior, and the other two altitudes coincide with the legs of the triangle.
- Acute triangle: all three altitudes lie inside the triangle.

**Where is the orthocenter located?**

- Obtuse triangle: the orthocenter is outside the triangle.
- Right triangle: the orthocenter coincides with the right angle’s vertex.
- Acute triangle: the orthocenter is an inner point.

## Altitude of an Equilateral Triangle

The **altitude** (*h*) of the **equilateral triangle** (or the height) can be calculated from Pythagorean theorem. The sides *a*, *a/2* and *h* form a right triangle. The sides *a/2* and *h* are the legs and *a* the hypotenuse.

Applying the Pythagorean theorem:

And we obtain that the **height** (*h*) of **equilateral triangle** is:

Another procedure to calculate its **height** would be from trigonometric ratios.

With respect to the angle of 60º, the ratio between altitude *h* and the hypotenuse of triangle *a* is equal to sine of 60º. Therefore:

## Altitude of an Isosceles Triangle

The **altitude** (*h*) of the **isosceles triangle** (or **height**) can be calculated from Pythagorean theorem. The sides *a*, *b/2* and *h* form a right triangle. The sides *b/2* and *h* are the legs and *a* the hypotenuse.

And it is obtained that the **height** *h* is:

In a **isosceles triangle**, the **height** corresponding to the base (*b*) is also the angle bisector, perpendicular bisector and median.

## Altitude of a Right Triangle

In a right triangle the **altitude** of each leg (*a* and *b*) is the corresponding opposite leg. Thus, h_{a} = *b* and h_{b} = *a*. The altitude of the hypotenuse is h_{c}.

The three altitudes of a triangle intersect at the orthocenter *H* which for a right triangle is in the vertex *C* of the right angle.

To find the height associated with side *c* (the hypotenuse) we use the geometric mean altitude theorem.

We can calculate the altitude *h* (or h_{c}) if we know the three sides of the right triangle.

Download this **calculator** to get the results of the formulas on this page. Choose the initial data and enter it in the upper left box. For results, press ENTER.

Triangle-total.rar or Triangle-total.exe

Note. Courtesy of the author: **José María Pareja Marcano**. Chemist. Seville, Spain.

## Exercise

Find the lengths of the three **altitudes**, h_{a}, h_{b} and h_{c}, of the triangle Δ *ABC*, if you know the lengths of the three sides: *a*=3 cm, *b*=4 cm and *c*=4.5 cm.

Firstly, we calculate the **semiperimeter** (*s*).

We get that semiperimeter is *s* = 5.75 cm. Then we can find the altitudes:

The lengths of three **altitudes** will be ** h_{a}=3.92 cm**,

**and**

*h*=2.94 cm_{b}**.**

*h*=2.61 cm_{c}