**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.

- Figure 5 below represents different Convex Polygon(a) and Concave Polygon(b)

**Figure 5: Convex and Concave polygon **

**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 * 4****2**

**= ****28****2**

**=** 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**

- 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.
- 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* 11****2**

**=****154****2**** = 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 * 7****2**** **

**=****70****2**** = 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 * 4****2**

**=****28****2**** = 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 * 0****2**** **

** = 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.