Altitude of a Triangle — Definition, Formula, How to Find & Examples

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

What is the altitude of a triangle?

An altitude is a line drawn from a triangle's vertex down to the opposite base, so that the constructed line is perpendicular to the base. Imagine you ran a business making and sending out triangles, and each had to be put in a rectangular cardboard shipping carton. How big a rectangular box would you need? Your triangle has length, but what is its height?

The height or altitude of a triangle depends on which base you use for a measurement. Here is scalene $\triangle GUD$. We can construct three different altitudes, one from each vertex.

Draw a scalene △GUD with ∠G=154°, ∠U=14.8°, and ∠D=11.8°. Label the sides too; side GU=17 cm, UD=37 cm, and DG=21 cm.

For $\triangle GUD$, no two sides are equal and one angle is greater than 90°, so you know you have a scalene, obtuse (oblique) triangle.

The altitude from $\angle G$ drops down and is perpendicular to UD, but what about the altitude for $\angle U$?

To get that altitude, you need to project a line from side DG out very far past the left of the triangle itself.

To get the altitude for $\angle D$, you must extend the side GU far past the triangle and construct the altitude far to the right of the triangle.

Classifying triangles

To find the altitude, we first need to know what kind of triangle we are dealing with. A triangle gets its name from its three interior angles. You can classify triangles either by their sides or their angles. By their sides, you can break them down like this:

Sides

• Scalene - No two sides are congruent (equal in length)

• Isosceles - Two sides are congruent

• Equilateral - Three sides are congruent

Most mathematicians agree that the classic equilateral triangle can also be considered an isosceles triangle, because an equilateral triangle has two congruent sides.

Angles

By their interior angles, triangles have other classifications:

• Right - One right angle (90°) and two acute angles

• Oblique - No right angles

Oblique Triangles

Oblique triangles break down into two types:

• Acute triangles - All interior angles are acute, or each less than 90°

• Obtuse triangles - One interior angle is obtuse, or greater than 90°

How to find the altitude of a triangle

Every triangle has three altitudes. Think of building and packing triangles again. You would naturally pick the altitude or height that allowed you to ship your triangle in the smallest rectangular carton, so you could stack a lot on a shelf.

Altitude for side UD ($\angle G$) is only 4.3 cm. What about the other two altitudes? If you insisted on using side GU ($\angle D$) for the altitude, you would need a box 9.37 cm tall, and if you rotated the triangle to use side DG ($\angle U$), your altitude there is 7.56 cm tall.

Not every triangle is as fussy as a scalene, obtuse triangle.

Altitude of a triangle example

What about an equilateral triangle, with three congruent sides and three congruent angles. Imagine $\triangle EQU$ with sides labeled 24 yards.

It will have three congruent altitudes, so no matter which direction you put that in a shipping box, it will fit. You only need to know its altitude.

Constructing an altitude from any base divides the equilateral triangle into two right triangles, each one of which has a hypotenuse equal to the original equilateral triangle's side, and a leg $\frac{1}{2}$ that length.

The other leg of the right triangle is the altitude of the equilateral triangle, so solve using the Pythagorean Theorem:

Anytime you can construct an altitude that cuts your original triangle into two right triangles, Pythagoras will do the trick! Use the Pythagorean Theorem for finding all altitudes of all equilateral and isosceles triangles.

For right triangles, two of the altitudes of a right triangle are the legs themselves. But what about the third altitude of a right triangle?

Picture right $\triangle RYT$ with the hypotenuse stretching horizontally.

Can you see how constructing an altitude from $\angle R$ down to side YT will divide the original, big right triangle into two smaller right triangles? Use Pythagoras again!

Where to look for the altitude

Where to look for altitudes depends on the classification of triangle. Since every triangle can be classified by its sides or angles, try focusing on the angles:

1. For oblique, acute triangles: the altitudes will always be inside the triangle

2. For oblique, obtuse triangles: the altitude dropped from the obtuse angle will be inside the triangle and the other two altitudes will fall outside the triangle

3. For right triangles, two of the altitudes are the legs and the third altitude is inside the triangle

Lesson summary

Now that you have worked through this lesson, you are able to recognize and name the different types of triangles based on their sides and angles. You now can locate the three altitudes of every type of triangle if they are already drawn for you, or you can construct altitudes for every type of triangle. For equilateral, isosceles, and right triangles, you can use the Pythagorean Theorem to calculate all their altitudes.