Education

# DIAGONAL OF A POLYGON- FACTS AND FORMULA

## INTRODUCTION

In and around you, you must have encountered many shapes. They may be closed, regular, irregular etc. These shapes can have an equal or unequal number of sides. They are commonly called polygons.

“ A polygon is a fully closed, flat, two-dimensional (2D) form with straight sides (all the sides are joined up). Straight sides are required. Polygons can have as many sides as they want. “

Fig 1: A polygon                    Fig 2: A closed with Fig 3: A shape that is not

Curved side is not a                       fully closed is also

Polygon                                           not a polygon

## REGULAR AND IRREGULAR POLYGONS

• Regular Polygon: A regular polygon has all of its sides the same length and angles the same.
•  Irregular Polygon: An irregular polygon has different lengths and widths of sides and angles.

Below are a few examples of some regular and irregular polygons.

### Fig 4: Regular and Irregular Polygon

Below is the list of a different regular polygon and their sides in a tabular form

 Sl.No Name of the Polygon No of sides 1 Pentagon 5 2 Hexagon 6 3 Heptagon 7 4 Octagon 8 5 Nenogon 9 6 Decagon 10

### Polygons are again divided into two-types

• Convex Polygon
• Concave Polygon
• Convex Polygon : If the interior angle of any enclosed shape is less than 180 degrees it is a Convex Polygon. All regular polygons are convex.
• Concave Polygon: If at least one interior angle of an enclosed shape is greater than 180 degrees, it is a Concave Polygon.
1. Figure 5  below represents different Convex Polygon(a) and Concave Polygon(b)

## WHAT IS THE DEFINITION OF DIAGONAL?

A diagonal is a line segment that runs through the vertex of a polygon and connects its opposing corners. A vertex is a shape’s corner. Line segments connecting non-adjacent vertex points of a polygon are called diagonals. It connects the vertices of a polygon, ignoring the figure’s edges. A diagonal has been painted on the following shapes:

### Diagonal of a Polygon – Formula

We can calculate the number of diagonals of any polygon with the help of a formula.

Suppose if we know the number of sides, we can calculate the number of diagonals and vice versa. Calculating the number of polygon diagonals is made easy with the Diagonal of a

### Polygon – Formula

We can simply compute the number of diagonals in a polygon using this formula.

The formula to calculate the diagonal is given below.

“ n” is the number of sides of a given polygon.

 Total no. of diagonals = n (n-3)2; n being the no. of sides

Example 1: Suppose you want to find the no of diagonals of a heptagon

Since you know the shape is a heptagon, so n=7

Using the formula, you can find the number of diagonals.

= n (n-3)2

= 7 * 42

= 282

= 14

### Therefore Heptagon has 7 sides and 14 diagonals.

If you know the number of diagonals, you can find the polygon using the same diagonal formula.

Example 2: If the number of diagonal is 35, find the number of sides of a polygon

No. of Diagonal =n (n-3)2

35 = n (n-3)2

35 * 2 = n(n-3)

n(n-3)=70

n2 – 3n – 70=0

n2 – 10n + 7n – 70=0

n( n- 10) + 7(n-10) = 0

n+ 7 =0 or n-10=0

n= -7  or n= 10

The number of sides cannot be negative, so n = 10

### The number of sides is 10, and it is a decagon

The following table displays the number of sides and the number of diagonals in various polygons:

 sl.no Name of the Shape No of diagonals 1 Triangle 0 2 Quadrilateral 2 3 Pentagon 5 4 Hexagon 9 5 Heptagon 14 6 Octagon 20 7 Nenogon 27 8 Decagon 35

## LENGTH OF THE DIAGONAL OF A SQUARE AND RECTANGLE

• SQUARE

Consider a square with side a

Using Pythagorean Theorem

d2 = a2 + a2

d2 = 2 a2

d=2a

 Diagonal of Square = 2a

RECTANGLE

Using Pythagorean Theorem

d2 = a2 + b2

d= a2 + b2

 Diagonal of a Rectangle = a2 + b2

(3) DIAGONAL OF A CUBE FORMULA:

By using the Pythagorean Theorem, we can determine the diagonal of any cube.

 Diagonal of a Cube= ( s2 + s2+ s2); where s is the side of the cube

The Use of Diagonals in Our Everyday Lives

1. Whether it’s a residential wall, a bridge, or a towering structure, diagonals in rectangles and squares add strength to a building’s construction. You may have observed that diagonal wires stabilise the bridges. To maintain the door straight and true, look for diagonal bracing in the structure.
2. This course’s diagonal is measured on the phone or computer screen you’re seeing it on. A 21″ screen’s width and height are never defined; it is 21″ from one corner to the other.

SOLVED EXAMPLES

EXAMPLE 1: Determine the total number of diagonals in a 14-sided regular polygon.

Solution:

It’s a 14-sided polygon means the number of sides is 14 and the vertices

n=14

So now, the number of diagonal can be calculated

=n (n-3)2

=14 (14-3)2

= 14* 112

=1542 = 77

Answer: 14 sided polygon will contain  77 diagonals

EXAMPLE  2

If a polygon has 90 diagonals, calculate the number of sides.

Solution:

Let the number of sides of this polygon is “ n”

Number of diagonals given = 90

Using the diagonal of a polygon formula,

Number of diagonals  =n (n-3)2

90 = n (n-3)2

n (n-3) = 180

n2– 3n =180

n2 – 3n – 180=0

n2 – 15n + 12n -180 = 0

n(n-15) +12(n – 15) = 0

(n + 12) (n – 15)=0

So n = – 12 or n = 15

Since the number of sides cannot be negative, the value of n is 15

Answer: The polygon with 90 diagonals have 15 sides

EXAMPLE 3:

Calculate the number of diagonals of a decagon

Solution

We know that the polygon is a decagon

So the number of sides n = 10

Using the diagonal of a polygon formula,

Number of diagonals  =n (n-3)2

=n (n-3)2

=10 (10-3)2

= 10 * 72

=702 = 35

Answer: A Decagon has 10 sides and 35 diagonals

EXAMPLE  4

Calculate the number of diagonals of a heptagon

Solution

We know that the polygon is a heptagon

So the number of sides n = 7

Using the diagonal of a polygon formula,

Number of diagonals  =n (n-3)2

=n (n-3)2

=7 (7 – 3)2

= 7 * 42

=282 = 14

Answer: A Heptagon has 7 sides and 14 diagonals

EXAMPLE 5

Calculate the number of diagonals of a triangle

Solution

We know that the polygon is a triangle

So the number of sides n = 3

Using the diagonal of a polygon formula,

Number of diagonals  =n (n-3)2

=n (n-3)2

=3 (3 – 3)2

= 3 * 02

= 0

Answer: A triangle has 3 sides and zero diagonals

CONCLUSION

We’ve learned a lot about really important polygon diagonals. As a result, we now know how to calculate the diagonals of any polygon and how to apply the method to several real-world examples. In addition, we discussed how diagonal formulae could be used to find the diagonal length in cubes, squares, and rectangles.

### Thomas Navarra 