# Polygons

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## Define Polygons

Thinking of what polygons are? In Euclidean geometry, a polygon is a crucial concept. Polygons are a closed plane figure in which all the sides are line segments. Each side should bisect exactly two other sides however only at their endpoints. The side of polygons should be noncollinear and must have a common endpoint. Having said that, there are different types of polygons and each polygon is generally named considering the number of sides it has. Note that a polygon having n-sides (=12) is called an n-gon.

### Polygon Examples

Classic daily life polygon examples include the building which houses the US Department of Defense called pentagon because it has 5 sides.

Below is a clear depiction of polygons along with sides and illustrations of each.

## Polygon Examples With Sides

 Names of Polygon Number of Sides Illustration Triangle 3 (Image will be uploaded soon) Quadrilateral 4 (Image will be uploaded soon) Pentagon 5 (Image will be uploaded soon) Hexagon 6 (Image will be uploaded soon) Heptagon 7 (Image will be uploaded soon) Octagon 8 (Image will be uploaded soon) Nonagon 9 (Image will be uploaded soon) Decagon 10 (Image will be uploaded soon)

### Types of Polygon

Based upon the sides and angles of a closed plane figure, we can classify the polygons into following different types, namely:

1. Regular Polygon

2. Irregular Polygon

3. Concave polygon

4. Convex Polygon

### Types of Polygons and Its Properties

1. Regular Polygon

If all the sides and the interior angles of the polygon measure equal, then it is what we call a regular polygon. A regular polygon is a polygon which has all the sides and all the angles congruent. The examples of regular polygons include plane figures such as square, rhombus, equilateral triangle, etc.

1. Irregular Polygon

In case of all the sides and the interior angles of the polygon do not measure similar, then it is called as an irregular polygon. Examples of irregular polygons include a rectangle, a kite, and a scalene triangle etc.

1. Concave Polygon

If one or more interior angles of a polygon measure more than 180Â°, then it is referred to as a concave polygon. A concave polygon can consist of a minimum of four sides. The vertex will point towards the interior of the polygon.

1. Convex Polygon

If all the interior angles of a polygon strictly measure less than 180Â°, then it is called a convex polygon. The vertex points outwards from the centermost of the shape.

### Properties of Polygons

As already mentioned, polygons are classified depending upon their sides, shape, angle and properties. Thus, here we bring you the key properties of polygons that will help you to simply determine the types of polygons very easily.

• Sum of all the interior angles of an n-sided polygon = (n â€“ 2) Ã— 180Â°.

• The angle measurement of each interior angle of an n-sided regular polygon = [(n â€“ 2) Ã— 180Â°]/n.

• The angle measurement of each exterior angle of an n-sided regular polygon = 360Â°/n.

• Number of diagonals in a polygon having n sides = n(n â€“ 3)/2.

• Number of triangles formed by connecting the diagonals from one corner of a polygon = n â€“ 2.

### Introduction to Angles of Polygons

As we are already familiar, any polygon consists of as many vertices as its sides. In addition, each corner possesses a specific measure of angles. These angles are classified into two kinds of angles namely interior angles and exterior angles of a polygon.

### Interior Angle Property of a Polygon

If we need to calculate the sum of all the interior angles of a simple n-gon, then we use the formula = (n âˆ’ 2) Ã— 180Â°.

Or

Sum of all the interior angles formula = (n âˆ’ 2) Ï€ radians.

In which â€˜nâ€™ is equivalent to the number of sides of a polygon.

For instance, a quadrilateral consists of four sides, thus, the sum of all the interior angle is given as:

Sum of interior angles of a 4-sided polygon i.e. a quadrilateral= (4 â€“ 2) Ã— 180Â°

= 2 Ã— 180Â°

= 360Â°

### Exterior Angle Property of a Polygon

This property postulates that the sum of the interior, as well as the corresponding exterior angles at each vertex of any polygon, is supplementary to each other, meaning that will be equal to 180 degrees. That being said, for a polygon;

Interior angle + Exterior angle = 180Â°

Exterior angle = 180Â° â€“ Interior angle

### Area and Perimeter Formulas of Polygons

The area and perimeter of different types of polygons are dependent on the sides.

Area of a polygon is described as the region covered by the particular polygon in a 2D plane.

The perimeter of a polygon is defined as the total distance covered by the sides of a polygon.

### Fun Facts

There is also a seamless polygon triangle.

Q1: How to Find the Total Number of Angle Measurement of a 5-Sided Polygon?

Answer: There is a useful formula to calculate the total (or sum) of internal angles for any polygon.

(n - 2) Ã— 180Â° (n = number of sides). To say, we need to find the angle of a 5-sided polygon i.e pentagon, the calculation would be:

5 - 2 = 3

3 Ã— 180 = 540Â°.

Thus, the sum of internal angles for any pentagon (excluding complex) = 540Â°.

Q2: How to Find Angle Measurement of a 5-Sided Polygon if it is Regular?

Answer: If the shape of a polygon is regular (all lengths of sides and angles are equal) then we would have to simply divide the sum of the internal angles by the total number of sides in order to find each internal angle i.e

540 Ã· 5 = 108Â°.

A regular pentagon thus contains five angles each of 108Â°.

Q3: How do we Calculate the Area of any Regular Polygon?

Answer: The easiest way to calculate the area of any regular polygon is to simply divide it into triangles, and apply the formula for the area of a triangle. For example, to calculate the area of a hexagon polygon, we will divide it into triangles (six).

The area is Height (depicted in red line) Ã— length of the side (depicted in blue line) Ã— 0.5 Ã— 6 (since there are 6 triangles).