Definition: Altitude of a triangle is the perpendicular drawn from the vertex of the triangle to the opposite side. Also, known as the height of the triangle, the altitude makes a right angle triangle with the base. Below is an image which shows a triangle’s altitude.
The main use of the altitude is that it is used for area calculation of the triangle i.e. area of a triangle is (½ base × height). Now, using the area of a triangle and its height, the base can be easily calculated as Base = [(2 × Area)/Height]
About altitude, different triangles have different types of altitude. Below is an overview of different types of altitudes in different triangles.
For an obtuse-angled triangle, the altitude is outside the triangle. For such triangles, the base is extended and then a perpendicular is drawn from the opposite vertex to the base. For an obtuse triangle, the altitude is shown in the triangle below.
The altitude or height of an equilateral triangle is the line segment from a vertex that is perpendicular to the opposite side. It is interesting to note that the altitude of an equilateral triangle bisects its base and the opposite angle. The image below shows an equilateral triangle ABC where “BD” is the height (h), AB = BC = AC, ∠ABD = ∠CDB, and AD = CD.
The altitude of a right-angled triangle divides the existing triangle into two similar triangles. According to right triangle altitude theorem, the altitude on the hypotenuse is equal to the geometric mean of line segments formed by altitude on hypotenuse.
The isosceles triangle altitude bisects the angle of the vertex and bisects the base. It should be noted that an isosceles triangle is a triangle with two congruent sides and so, the altitude bisects the base and vertex.
|Triangle Type||Altitude Formula|
|Equilateral Triangle||h = (½) × √3 × s|
|Isosceles Triangle||h =√(a2−b2⁄4)|
|Right Triangle||h =√(xy)|
For an equilateral triangle , all sides are equal to 60°.
So, sin 60° = h/AS
We know, AS = AC/2 or s/2 (since all sides are equal)
∴ sin 60° = [h/(s/2)]
⇒ Altitude of an equilateral triangle = h = √(3⁄2) × s
To calculate the area of a right triangle, the right triangle altitude theorem is used which has been explained in the linked article in detail. Also, get the complete derivation for the altitude of a right triangle from the linked article.
⇒ Altitude of a right triangle = h = √xy