In this post we will discuss about polygons, its properties and different formulas involved for calculation of various parameters of the polygons. I have tried to develop this content as comprehensive as possible for students of Grade 6.

## Polygon Properties & Types

### What is a polygon??

If a simple closed figure is made up entirely of line segments, then it is called polygon.

Remember that the figures are closed meaning it has no beginning and end points. Also each polygon has an enclosed area which can be calculated with the help of different techniques

**Properties of polygon **

**(1) **Common naming of polygon

In general, if a polygon has n sides, it is named as “n-gon”.

**Ex –**

If a polygon has 16 sides, it is called 16-gon

If a polygon has 19 sides, it is called 19-gon

Name of Polygon | Number of Sides |

Triangle | 3 |

Quadrilateral | 4 |

Pentagon | 5 |

Hexagon | 6 |

Heptagon | 7 |

Octagon | 8 |

Nonagon | 9 |

Decagon | `10 |

#### 2. Interior Angle Property of Polygon

The sum of all the interior angles of a simple n-gon = (n − 2) × 180°

Or

Sum = (n − 2) x π radians

Where ‘n’ is equal to the number of sides of a polygon.

**Ex Question–**

A polygon has four sides, then, sum of all the interior angle is given by:

Sum of interior angles of 4-sided polygon = (4 – 2) × 180°

= 2 × 180°

= 360°

Here, all the above figures are 4-sided polygon and the sum of all interior angles of each polygon is equal to 360°.

**Example Question 02**

A polygon has 6 sides, then, sum of all the interior angle is given by:

Sum of interior angles of 6-sided polygon = (6 – 2) × 180°

= 4 × 180°

= 720°

you can see all the interior angles of hexagon in the above figure, you can easily calculate that all the interior angle of hexagon adds up to 720 degree

**Category of Question asked from Interior angle of Polygon**

**Category 01****(01) If a sum of all interior angles of a polygon is 1080 ^{o}, then find the number of sides in the polygon**.

Sum of all the interior angles of a simple n-gon = (n − 2) × 180°

1080^{o} = (n-2) x 180^{o}

^{ }1080^{o}/180^{o} = n – 2

6 = n – 2

6 + 2 = n

8 = n

Therefore, the polygon has 8 sides

**Category 02**

Find the measure of each internal angle of a 6-sided regular polygon

Sum of all the interior angles of a simple n-gon = (n − 2) × 180°

Here, n = 6

= (6 – 2) x 180°

= 4 x 180^{o}

= 720^{o}

Measure of each interior angle

= 720^{o}/6

= 120^{o}

**Category 03**

Find the value of x for the given figure

Sol. It is a 6-sided polygon

Hence, sum of interior angles = (6 – 2) x 180^{o}

120^{o} + 90^{o} + 110^{o} + 130^{o} + 160^{o} + x^{o} = 4 x 180^{o}

610^{o} + x^{o} = 720^{o}

x^{o} = 720^{o} – 610^{o}

x^{o} = 110^{o}

#### 03. Interior and Exterior Angle

The sum of interior and the corresponding exterior angles at each vertex of any polygon are supplementary to each other. i.e. for a polygon –

Interior angle + Exterior angle = 180^{o}

Or Interior angle = 180^{o} – Exterior angle

** 04: **Number of Diagonal of Polygon

The number of diagonals in a polygon with n sides = n x (n – 3)/2

Here, the polygon has total 5 sides

Therefore, no. of diagonals formed in the polygon are = n x (n – 3)/2

=> 5 x (5 – 3)/2

=> 5 x 2/2

=> 5

Therefore, 5 diagonals are formed

** 05: Number of Triangle in Polygon**

The number of triangles formed by joining the diagonals from one corner of a polygon = n – 2

Here, the polygon has total 5 sides

Therefore, no. of triangles formed by joining the diagonals from one corner of the polygon = n – 2

= 5 – 2

= 3

Therefore, 3 triangles are formed

**06: **Interior Angle calculation of Polygon

The measure of each interior angle of n-sided regular polygon = ** [(n – 2) × 180°]/n**

Here, the polygon has total 5 sides

Therefore, measure of each interior angle is = [(n – 2) × 180°]/n

=> [(5 – 2) × 180°]/5

=> [3× 180°]/5

=> 540°/5 => 108^{o}

** 07: **Exterior Angle Formula for Polygon

The measure of each exterior angle of an n-sided regular polygon = 360°/n

Here, the polygon has total 5 sides

Therefore, measure of each exterior angle is = 360°/n

= 360°/5

=72°

### Types of Polygon

Different types of Polygons are

1.Regular Polygon

2. Irregular Polygon

3. Convex Polygon

4. Concave Polygon

**Regular polygon**

A polygon with all its sides and angles equal are called regular polygon.

Ex – regular pentagon, regular hexagon etc.

**Irregular polygon **

A polygon with unequal sides and angles are called regular polygon.

Ex- Irregular triangle, irregular pentagon etc.

**Convex Polygon **

- If all the interior angles of a polygon are strictly less than 180 degrees, then it is known as a convex polygon.
- The vertex will point outwards from the center of the shape.
- Diagonals lie inside the polygon

Here, all the interior angles are less than 180^{o}

All the vertex A, B, C, D and E points outward from the center of the shape & all the 5 diagonals AD, AC, BD, BE and CE are inside the polygon

So, the polygon is a convex polygon

**Concave Polygon **

- If one or more interior angles of a polygon are more than 180 degrees, then it is known as a concave polygon.
- A concave polygon can have at least four sides.
- The vertex points towards the inside of the polygon
- one or more diagonal lie outside the polygon

Here, one interior angle is more than 180^{o }(i.e. CDE)

Vertex D points inward from the center of the shape

Diagonal CE lies outside the polygon

So, the polygon is a concave polygon