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
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 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.
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, ha = b and hb = a. The altitude of the hypotenuse is hc.
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 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.
Triangle-total.rar or Triangle-total.exe
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