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# Diagonal|Definition & Meaning

## Definition

A line segment joining the two vertices or corners of the non-adjacent sides of a polygon is known as a diagonal. The corners must be opposite to each other for a diagonal. It is not a part i.e. side of a polygon.

**Figure 1** shows the demonstration of **diagonals** in different colors. The line segments **AB**, **CD**, and **EF** are the diagonals in the **square, rectangle,** and **hexagon** respectively.

## Historical Background

The word **“diagonal”** comes from the **Greek** word **“diagonios”** which means “from one angle to another”. **Euclid** and **Strabo** referred to it as a line joining two opposite vertices of a **geometrical** shape.

It also comes from the **Latin** word **“diagonus”** which means **“inclining** line”. So from the ancient perspective, a diagonal is a line segment having some **slope** titled at some angle connecting **non-adjacent** vertices.

## Geometry

Geometry is the **field** of mathematics that involves the study of different **shapes** and their geometrical interpretations and **calculations.** Diagonals were recognized in the **early** developing stages of geometry.

The following **terms** play an essential role while understanding the concept of **diagonal.**

## Polygon

A polygon is a **geometrical** shape consisting of **three** or more lines. It is a **two-dimensional** structure whose boundaries are the line segments connected to form a **closed** figure.

“**Poly**” means “**many**” so there can be as many line segments to form its boundary. **Figure 2** shows different types of polygons.

### Non-adjacent Vertices

The points which join the ends of the two **sides** of a polygon are known as the **vertices.** Two sides having a common vertex are known as **adjacent** sides.

**Non-adjacent** vertices always form diagonals. These are the **vertices** that don’t share a common side. They are **opposite** to each other and are not joined by a common edge.

**Figure 3** shows the non-adjacent vertices in some polygons.

### Edges

Edges are the **sides** of a polygon. The **adjacent** vertices are connected by a line segment **that forms a polygon’s edges.** The diagonal is not one of these edges.

## Concave Polygons

Concave polygons are such polygons that have at least one **internal angle greater** than **180°**. The two sides are bent inwards. **Figure 4** shows a concave pentagon.

### Diagonals of Concave Polygon

A **concave** polygon has a diagonal **outside** its boundary when drawn from the concave non-adjacent **vertices.**

**Figure 5** shows a **diagonal** in orange color which is outside the **concave** pentagon.

## Number of Diagonals in Different Polygons

The **number** of **diagonals** of a polygon can be calculated by using the **formula** given below:

No. of Diagonals = n = p(p – 3)/ 2

Where “**p**” is the number of** sides** or **edges** of the polygon. The number of diagonals of the following **polygons** is calculated.

### Triangle

A triangle has **no diagonal** as the **non-adjacent** vertices in a triangle are already joined by its **edges.** This can be proved by using the **diagonal** formula.

A triangle has **three** sides, so **p** will be **3**. Putting the value in the above equation gives:

No. of Diagonals in a Triangle = n = 3(3 – 3)/ 2

**n = 0**

The **diagonals** are present in the polygons with sides **greater** than **three.**

### Quadrilateral

For a quadrilateral,** p = 4**, so the number of **diagonals** in a quadrilateral will be:

n = p(p – 3)/ 2

n = 4(4 – 3)/ 2

**n = 2**

So, **two** diagonals can be formed in a **quadrilateral.**

### Pentagon

A pentagon has five edges, so** p = 5**, the number of **diagonals** will be:

n = p(p – 3)/ 2

n = 5(5 – 3)/ 2

**n = 5**

Hence, a **pentagon** has **five** diagonals.

### Hexagon

For hexagon,

No. of sides =** p = 6**

The number of **diagonals** will be:

n = p(p – 3)/ 2

n = 6(6 – 3)/ 2

**n = 9**

So, there are **nine** diagonals in a **hexagon.**

### Heptagon

A heptagon has seven sides, so **p = 7**. The number of **diagonals** in a heptagon will be:

n = p(p – 3)/ 2

n = 7(7 – 3)/ 2

**n = 14**

So, a heptagon has **fourteen** diagonals.

**Figure 6** shows the diagonals in these polygons.

## Length Calculation of Diagonals in Different Polygons

The **diagonal length** can be calculated if we know the lengths of the sides in different polygons. Following are some **polygons** whose length **formulas** for diagonals are discussed.

### Square

A square has **equal sides. **Let **m** be the **length** of a square **side;** the **diagonal length** can be calculated with the **formula:**

Length of Diagonal = d = m$\sqrt{2}$

### Rectangle

A rectangle has two sides **longer** than the other two sides. Let **r** be the **length** and **s** be the **breadth** of the rectangle. The **diagonal** length in a rectangle is calculated by using the **formula**:

Length of Diagonal = d = $\sqrt{ r^2 + s^2 }$

Bisection means dividing into **two** equal parts. The **diagonals** of **quadrilaterals**(square, rectangle, parallelogram) always **bisect** them.

## Solved Examples of Polygons Involving Diagonals

### Example 1

Find the number of diagonals in an **octagon** and a **decagon.**

### Solution

“**Octa**” in the octagon means “**eight**”. So, an octagon has eight **sides.** The **formula** for the number of diagonals is:

No. of Diagonals = n = p(p – 3)/ 2

Putting **p = 8** in the above equation gives:

**No. of Diagonals = n = 8(8 – 3)/ 2**

n = 8(5)/2

n = 40/2

**n = 20**

Hence, the number of **diagonals** in an octagon will be **twenty**.

A **decagon** has ten sides as “**deca**” means “**ten**”, so p = 10. The number of diagonals will be:

**No. of Diagonals = n = 10(10 – 3)/ 2**

n = 10(7)/2

n = 70/2

**n = 35**

So,** thirty-five** diagonals can be drawn in a **decagon.**

### Example 2

A **rectangle** has a length and width of **18 cm** and **9 cm**. What is the length of the **diagonal** in this rectangle?

### Solution

The formula to calculate the **diagonal’s length** in a **rectangle** is given by:

Length of Diagonal = d = $\sqrt{ r^2 + s^2 }$

Where** r** is the **length** and **s** is the width or **breadth**. The values of **r** and **s** for this rectangle will be:

r = 18 cm , s = 9 cm

Putting the **values** in the equation gives:

Length of Diagonal = d = $\sqrt{ {18}^2 + {9}^2 }$

d = $\sqrt{ 324 + 81 }$

d = $\sqrt{ 405 }$

**d = 20.1 cm**

So, the **length** of the **diagonal** of the rectangle will be** 20.1 cm**.

*All the images are created using GeoGebra.*