Altitude of Triangle – Definition, Formulas, Examples, Properties

Last Updated : 28 Feb, 2024

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.

Altitude is a measurement in a triangle.

It is the distance from a vertex to the line containing the opposite side.

It is perpendicular to the opposite side.

Altitude represents the height of the triangle from a specific vertex to the base.

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.

Name of Triangle	Formula for Altitude
Right Angle Triangle	h = (Base × Height) / Hypotenuse
Equilateral Triangle	h = (Side × √3) / 2
Obtuse Triangle	h = (Base × Height) / (2 × Sin(angle))
Isosceles Triangle	h = √(Leg² – (Base / 2)²)
Scalene Triangle	Heron’s Formula:- Area = √(s(s – a)(s – b)(s – c))
Acute Triangle	h= (2 × Area of Triangle) / Base length

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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:

Key Factors	Median	Altitude
Definition	A vertical line segment drawn from the vertex to the midpoint of a triangle	A perpendicular line segment drawn from the vertex to the base of a triangle
Construction	Mostly constructed inside the triangle	The construction depends on the type of triangle i.e. maybe outside or inside.
Intersection Point	The meeting point of medians is called centroid.	The meeting point of medians is called orthocenter.
Relationship with Area	Bisects a triangle as well as its base into 2 equal sides.	It does not bisects a triangle as well as its base into 2 equal sides.

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.

FAQs on Altitude of a Triangle

1. What is altitude in a triangle in Geometry?

In geometry, the altitude of a triangle refers to the perpendicular distance from a vertex of the triangle to the line containing the opposite side. It is a line segment that forms a right angle with the base, creating a right-angled triangle with the base.

2. How to find Altitude of a Triangle?

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

Altitude = Area of the Traingle / Length of the corresponding base

If the length of the side of triangle are known then Pythagoras Theorem can be used.

3. How to draw the altitude of a triangle?

To draw the altitude of a triangle, follow these steps:

Select a vertex of the triangle.

Draw a line segment from the chosen vertex to the line containing the opposite side.

Ensure that the line segment forms a right angle with the base.

4. What is the Altitude of a Triangle Meaning?

The altitude of a triangle represents the perpendicular distance from a vertex to the opposite side. It plays a crucial role in determining the height of the triangle and is often used in various geometric calculations.

5. What is Median in Triangle?

A median in a triangle is a line segment connecting a vertex to the midpoint of the opposite side. A triangle has three medians, each originating from a different vertex.

6. Are Altitude of Triangle Concurrent?

Yes, the altitudes of a triangle are concurrent, meaning they intersect at a single point known as the orthocenter. The orthocenter is the point where all three altitudes of a triangle meet. This property holds true for all types of triangles—scalene, isosceles, or equilateral.

7. Can an Altitude be Outside the Triangle?

Yes, an altitude can be drawn outside the triangle from a vertex perpendicular to the opposite side depending on the type of triangle.

8. Do all triangles have altitudes?

Yes, all triangles have minimum one altitude, but some might have more than one, depending on their types.

9. What is the importance of the orthocenter in a triangle?

The orthocenter is the point where all three altitudes meet, reflecting properties like concurrency and triangle geometry.

10. Do we consider altitudes and medians same?

No, while both include line segments drawn from vertices, medians connect to midpoint, while altitudes connect perpendicularly with the opposite side and forms a right triangle.

11. How does the area of a triangle affect the altitude?

The altitude is used to compute he area of a triangle, as the area is equal to the product of the base and the corresponding altitude which at last divided by two.

12. Can an isosceles triangle contain more than one altitude?

Yes, an isosceles triangle can have two altitudes – one for each of the congruent sides & another one which intersects at the vertices opposite to the base.

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