Altitude of Triangle - Definition, Formulas, Examples, Properties

The Altitude of a triangle is the length of a straight line segment drawn from one of the triangle's vertices (corners) perpendicular to the opposite side.

It's like measuring the height of the triangle from a specific point to the base. The altitude is a fundamental concept in geometry and is often used to calculate the area of a triangle.

In this article, we have covered the Altitude of a Triangle, its Properties, the Altitude of each type of triangle, How to find Altitude, and many more in simple way.

Let's dive right in.

Altitude of a Triangle Definition

The altitude of a triangle is the perpendicular distance from a vertex (corner) of the triangle to the line containing the opposite side. It represents the height of the triangle measured from a specific vertex to the base, forming a right angle with the base.

The altitude of the triangle is mainly located inside it, but in certain cases, it is also found outside of the triangle.

In simple words, Altitude of a Triangle is defined as the following.

1. Altitude is a measurement in a triangle.
2. It is the distance from a vertex to the line containing the opposite side.
3. It is perpendicular to the opposite side.
4. Altitude represents the height of the triangle from a specific vertex to the base.
5. It forms a right angle with the base.

Definition of Altitude of a Triangle

An altitude is a straight line which is drawn from the vertex to the opposite side of a triangle. It is the line segment made from the corner of the triangle to the other side which forms a 90° angle.

Altitude of a Triangle Properties

The following are the properties of Altitude of Triangle:

• Perpendicularity: An altitude is always perpendicular to its respective base.
• Intersection at a Point: In triangles, altitudes intersect at a common point which is called the orthocenter.
• Length Relationships: In isosceles triangles, the altitude divides the base and opposite angle, resulting in two congruent right triangles.
• Area Computation: The product of the base and altitude of a triangle and its area is divided by 2.

Orthocenter: Intersection of Altitudes of a Triangle

The orthocenter of a triangle is the point where all three altitudes (also known as perpendiculars) of the triangle intersect.

Note:

• In an acute-angled triangle, the orthocenter lies inside the triangle.
• In Right-angled triangle, the orthocenter lies at the vertex containing the right angle.
• In an obtuse-angled triangle, the orthocenter lies outside the triangle.

Altitude of Different Triangles

Altitude represents the shortest distance from a vertex to the line formed by the opposite side. In different types of triangles— acute, obtuse, or right-angled, the altitude is essential in determining the area of the triangle.

1. Altitude of a Scalene Triangle

• Altitude (h) = 2 * Area / Base length
• Triangle has all sides of different lengths.
• Use Heron's Formula to find Area: sqrt(s(s - a)(s - b)(s - c)).
• Altitude formula: h = 2 * Area / Base length.

• Formula: h=2√s(s−a)(s−b)(s−c)/b
• h: Altitude or height of the triangle.
• s: Semi-perimeter (a+b+c​)/2
• a,b,c: Lengths of the sides of the triangle.

2. Altitude of an Equilateral Triangle

• An equilateral triangle has all sides equal.
• The altitude splits the triangle into two congruent right-angled triangles.
• Using Pythagoras theorem, if s is the side length, then h=2√s(s−a)(s−b)(s−c)/b .
• Simplifying gives ℎ=√3/2 X s

• Formula: h = (Side × √3) / 2
• h: Altitude.
• Side length: Length of any side in the equilateral triangle.

3. Altitude of an Isosceles Triangle

Altitude (h) = sqrt(a^2 - 1/4 * b^2)

The following are the Derivation for Altitude of Obtuse Triangle:

• Triangle has two equal sides.
• Altitude is perpendicular bisector of the base.
• Using Pythagoras theorem in △ADB: h^2 = a^2 - (1/2 * b)^2.
• Simplifying gives h = sqrt(a^2 - 1/4 * b^2).

• Formula: ℎ= h = √(Leg² - (Base / 2)²)
• ℎh: Altitude.
• a,b: Equal lengths of the isosceles sides.
• c: Length of the unequal side.

4. Altitude of a Right-Angled Triangle

Altitude (h) = a * b / c

The following are the Derivation for Altitude of Right-Angled Triangle:

• One angle is 90 degrees.
• Altitude forms two similar triangles.
• If a, b, and c are sides, where c is the hypotenuse: h = a * b / c.

• Formula: ℎ=h = (Base × Height) / Hypotenuse​
• h: Altitude.
• a,b: Legs of the right-angled triangle.
• c: Hypotenuse.

5. Altitude of an Acute Triangle

Altitude (h) = 2 * Area / Base length

The following are the Derivation for Altitude of Acute Triangle:

• Triangle has all angles < 90 degrees.
• Formula: h = 2 * Area / Base length.

• Formula: ℎ=2×Area of Triangle/Base length

6. Altitude Obtuse Triangle

Altitude (h) = Base * Height / (2 * sin(angle))

The following are the Derivation for Altitude of Obtuse Triangle:

• Triangle has one angle > 90 degrees.
• Altitude from the vertex opposite the obtuse angle.
• Formula: h = Base * Height / (2 * sin(angle)).

Formula: ℎ=Base×Height2×sin⁡(angle)h=2×sin(angle)Base×Height​

Understanding the formulas for altitudes in different types of triangles is crucial for exploring the geometric properties and relationships unique to each triangle.

How to Find Altitude of a Triangle?

To find the altitude of a triangle, you can use the formula:  

Altitude = Area of the Triangle \ Length of the Corresponding Base

If the lengths of the sides are known, the Pythagorean theorem can be applied to determine the altitude. The altitude represents the perpendicular distance from a vertex to the line containing the opposite side, providing valuable insights into the triangle's geometry and aiding in various calculations.

Median and Altitude of a Triangle

The concepts of medians and altitudes in geometry offer fundamental insights into the structural properties of triangles. Medians connect vertices to midpoints, while altitudes represent the perpendicular distances from vertices to opposite sides, providing a comprehensive understanding of triangle geometry.

Difference Between Median and Altitude of a Triangle

The following are the difference between Median and Altitude of a Triangle:

Solved Examples on Altitude of triangle

Problem 1: Find the altitude of a right triangle whose base is 6 units and height is 8 units.

Solution:

Use the formula of a right triangle's altitude: Altitude = (Base × Height) / Hypotenuse Altitude = (6 × 8) / Hypotenuse

Hypotenuse = 10 units (from Pythagorean theorem)

Thus, Altitude = (6 × 8) / 10 = 4.8 units

Problem 2: Determine the altitude of an equilateral triangle with a side length of 12 units.

Solution:

Using the formula for an equilateral triangle's altitude: Altitude = (Side × √3) / 2

Altitude = (12 × √3) / 2 = 6√3 units

Practice Questions on Altitude of a triangle

1. Determine the altitude of a right triangle whose base is 20 units and a height is 6 units.

2. Compute the altitude of an equilateral triangle whose one side length is 12 units.

3. Find the altitude of an obtuse triangle with a base of 16 units and height of 6 units.