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 (ha, hb and hc), 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
- Obtuse triangle: The altitude related to the longest side is inside the triangle (see hc, in the triangle above) the other two heights are outside the triangle (ha, and hb).
- 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.
Altitude of an Isosceles Triangle
And it is obtained that the height h is:
Altitude of a Right Triangle
We can calculate the altitude h (or hc) 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.
Note. Courtesy of the author: José María Pareja Marcano. Chemist. Seville, Spain.
Find the lengths of the three altitudes, ha, hb and hc, 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 ha=3.92 cm, hb=2.94 cm and hc=2.61 cm.
Lines Associated with a Triangle
AUTHOR: Bernat Requena Serra