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# Altitude (Geometry)|Definition & Meaning

## Definition

In mathematics (geometry to be specific), **altitude** is the **perpendicular line segment** connecting a **vertex** of a triangle to a point on its **opposite facing side** at right angle. A triangle has three altitudes. The concept is not limited to triangles only, it can be extended to any shape with ** odd number of vertices.**

In more general terms, altitude means **straight line distance** (or displacement) of a point on an object standing on the ground or an object flying in the air, such that the **line is perpendicular to the horizon**.

## Visual Illustrations of Altitude in Triangles

Following figures **illustrate** the concept of **altitude** for **different types of triangles**. Figure 1 shows the altitude of a right triangle.

A **right angled triangle** is such a triangle for which one of the angles is equal to **90 degrees** (called the** right angle**). Its sides are termed as **base** (b), **hypotenuse** (hyp) and **perpendicular** (perp). The **altitude** of such a triangle can be calculated using the following **formula:**

\[ Altitude \ = \ \sqrt{ (CD)(DA) } \]

Where **CD** and **DA** are line segments. **Figure 2** shows the altitude of a **Equilateral** triangle. An **equilateral triangle** is such a triangle for which **all three angles or three sides are equal.**

The **altitude** of such a triangle can be calculated using the following **formula:**

\[ \text{Altitude} \ = \ \dfrac{ \sqrt{ 3 } }{ 2 } ( \text{ length of any side } ) \]

**Figure 3** shows the altitude of an **Isosceles triangle**. An **Isosceles triangle** is one whose** two angles or two sides are equal**.

The **altitude** of such a triangle can be calculated using the following **formula:**

\[ \text{Altitude} \ = \ \sqrt{ a^2 \ – \ \dfrac{ 1 }{ 4 } b^2 }\]

Where a is the length of any of the sides that are equal while b is the length of the third side. **Figure 3** shows the altitude of a **Scalene type of triangle.** A **Scalene** triangle is such a triangle for which **all angles and sides are unique** and none is equal.

The **altitude** of such a triangle can be calculated using the following **formula:**

\[ \text{Altitude} \ = \ \dfrac{ 2 \sqrt{ s (s-a) (s-b) (s-c) } }{ b } \]

Where a, b & c are the **lengths** of sides, s is the **semi perimeter** and b is the side for which altitude is being calculated. **Semi perimeter** is defined as:

\[ s \ = \ \dfrac{ a + b + c }{ 2 } \]

## Significance of Altitude (Geometry)

A **perpendicular line** drawn from a triangle’s vertex to its opposite side is called the **altitude of a triangle**. Since a triangle has **three sides,** it may be drawn along with a maximum of **three altitudes** .

There are several types of applications of altitudes in the geometrical problems related to triangles. The chief application among all is that the **area of** **a triangle** is calculated using the altitude, also known as height, of the triangle, which is usually represented by the** letter symbol ‘h’**.

Since a triangle’s height is determined by drawing a perpendicular line from its vertex to its opposite side, the altitude term is sometimes **analogously referred** to as the **triangle’s heigh**t, especially since it forms a right-angle triangle with the base.

The **altitude** is a very **important** parameter when we talk about **triangles** in geometry. They help us in proving many essential **laws of geometry.** The term is widely used in many other domains of **mathematics and physics** but we will restrict our focus on the term in geometry domain.

## Different Forms of Altitudes in Geometry Perspective

Altitudes have different mathematical perspectives for different types of triangles. Some of the key properties of altitudes are listed below:

**a.** A triangle can have a **maximum of three altitudes** (one per side).

**b.** The altitude may intersect the opposing side **inside or outside the triangle** depending upon the type of triangle under observation.

**c.** By definition, the altitude always** creates a 90° angle** with the side directly across from it.

There is **another important term** that comes across the reader whenever we are using altitudes in numerical problems. The term is called **orthocenter.** In the context of triangles, an orthocenter is the location where the triangle’s three elevations connect. In other words, the** point of intersection of all three altitudes** of a triangle is called its orthocenter.

The **numerical calculation** of the altitude length highly depends on the **type of triangle**. The formulae are listed in the next section. Let us consider four types of triangles: the **Right Triangle, ****Equilateral Triangle**, **Isosceles Triangle,** and **Scalene Triangle.**

## Numerical Examples for Calculating Altitudes

The **mathematical process** of evaluating altitude of a triangle problems is explained with the help of **numerical examples** in this section. Here we suppose that we wanted to calculate the results of **following operations:**

(a) Find the length of altitude of a **right triangle** if it splits the hypotenuse line segment into two parts of lengths 3 and 4.

(b) Find the length of altitude of an **equilateral triangle** if the length of each side is equal to 10.

(c) Find the length of altitude of an **isosceles triangle** if the length of two sides is equal to 10 and the third side is 4.

(d) Find the length of altitude of a **Scalene **triangle if the length of its sides is 7, 8 and 9.

### Solutions

**(a) Right Triangle**

Given that CD = 3 and DA = 4, we can use the formula:

\[ \text{Altitude} \ = \ \sqrt{ ( 3 ) ( 4 ) } \ = \ \sqrt{ 12 } \]

**(b) Equilateral Triangle**

Given that a = b = c = 10, we can use the formula:

\[ \text{Altitude} \ = \ \dfrac{ \sqrt{ 3 } }{ 2 } ( 10 ) \ = \ 5 \sqrt{ 3 } \]

**(c) Isosceles Triangle**

Given that a = 10 and b = 4, we can use the formula:

\[ \text{Altitude} \ = \ \sqrt{ (10)^2 \ – \ \dfrac{ 1 }{ 4 } (4)^2 } \ = \ \sqrt{ 100 \ – \ 4 } \ = \ \sqrt{ 96 } \]

**(d) Scalene Triangle**

Given that a = 7, b = 8 and c = 9, we can use the formula:

\[ s \ = \ \dfrac{ 7 + 8 + 9 }{ 2 } \ = \ \dfrac{ 24 }{ 2 } \ = \ 12 \]

Once we have semi perimeter, we can find the altitude with the following formula:

\[ \text{Altitude} \ = \ \dfrac{ 2 \sqrt{ 12 ( 12 – 7 ) ( 12 – 8 ) ( 12 – 9 ) } }{ 8 } \]

\[ \text{Altitude} \ = \ \dfrac{ 2 \sqrt{ 12 ( 5 ) ( 4 ) ( 3 ) } }{ 8 } \ = \ 6.8 \]

*All mathematical drawings and images were created with GeoGebra.*