Polygon

In geometry, polygons are the fundamental shapes that form the framework for more complex structures and patterns. From a simple triangle in mathematics to structures in complex buildings in real life, polygons are everywhere. Here, we will explore every aspect of this simple yet important concept of geometry, along with its formulas and types. So, let’s begin!

1.0Polygon Definition

The word polygon comes from Greek words, where “poly” means many and “gon” means angles. Thus, it can be concluded that a polygon is simply a shape with many angles. It is a two-dimensional, closed geometric figure formed by the intersection of a finite number of straight line segments connected end-to-end. 

The line segments in the polygons are known as sides, while the points where these lines meet are called vertices. The enclosed area within the line segments forms the interior of polygons; on the other hand, the connecting sides create the boundary of the shape. 

Every Polygon possesses certain features or elements, such as diagonals, interior and exterior angles, etc. 

2.0Properties of Polygons

Every polygon, regardless of its shape and number of sides, follows certain characteristics and properties. These properties of polygons help define and classify these shapes systematically. Some of these properties are: 

• Closed shape: Every polygon is a closed shape, such that the sides of it make a continuous loop.
• Straight sides: In polygons, note that the edges are created by straight line segments, not curves.
• Number of sides, angles, and vertices: If a polygon has “n” number of sides, then the number of vertices and interior angles will also be “n”. 
• No crossing or overlapping sides: The line segments only intersect at their ends, meaning they neither cross nor overlap other sides. 

3.0Types of Polygons

Polygons can be classified into four broad categories, namely concave, convex, regular and irregular polygons. Each of these types of polygons helps recognise their unique feature and characteristics, which include:  

Convex Polygons: A convex polygon is a polygon where all the interior angles are smaller than 180°, and none of the vertices face inward. If you connect any two points within the polygon with a line, it will always be entirely inside the shape. Examples of convex polygons include equilateral triangles, rectangles, regular pentagons, etc. 

Concave Polygon: A concave polygon will have at least one interior angle measure greater than 180°, and at least one vertex will seem to "cave in" towards the inside. Some diagonals will lie outside the polygon. Examples of a concave polygon include an arrowhead-shaped quadrilateral, a star-shaped polygon, etc.  

Regular Polygons: A regular polygon consists of all its sides being equal in length and all interior angles being equal in measure. It is always convex in nature. This kind of polygon always has high symmetry and equal angles. Examples of regular polygons include a square, an equilateral triangle, etc. 

Irregular Polygons: An irregular polygon is not equal in all sides or angles. These polygons may be convex or concave based on the positioning of their sides and angles. Examples of irregular polygons are a scalene triangle, an L-shaped polygon, etc. 

4.0Polygon Chart

Polygons are often named according to the number of sides they contain. Let’s explore the following polygon chart to understand the nomenclature of these polygons: 

5.0Polygon Formulas

Despite having different properties, every polygon shares certain common polygon formulas for its different elements, which include: 

Interior and Exterior Angles of Polygons

Interior and exterior angles of polygons are the two most important elements of these shapes, each of which can be defined as: 

Interior Angles: 

An interior angle is the angle formed by the intersection of two adjacent sides inside the polygon. The sum of all the angles of any polygon can be calculated using the formula:  

$\text{Sum of Interior Angles}=(n-2)\times 180°$

In a regular polygon, the measure of each interior angle is calculated as: 

$\text{Interior Angle}=\frac{(n-2)\times 180°}{n}$

Here, n in both formulas is the number of sides of a polygon. 

Exterior Angle: 

An exterior angle is created when one side of the polygon is extended, and the angle is created between the extended side and the following side. The interior and its corresponding exterior angles of a polygon always add up to 180°, while the total of all exterior angles of any polygon always equals 360°.

The measure of each exterior angle in a regular polygon is calculated using the formula: 

$ExteriorAngle=\frac{360°}{n}$

Diagonals in a Polygon

A diagonal is a line connecting two non-adjacent vertices of a polygon. All polygons (with more than three sides) have diagonals. The formula to calculate the diagonals in a certain polygon is expressed as: 

$\text{Number of diagonals}=\frac{n(n-3)}{2}$

6.0Solved Examples of Polygon

Problem 1: The exterior angle of a regular polygon is 40°. How many sides does it have? Also, name and classify the polygon as convex/concave.

Solution: Given that the exterior of a regular polygon is 40°. 

Using the following formula; 

$\text{Exterior Angle}=\frac{360°}{n}$

$40°=\frac{360°}{n}$

$n=9$

As the polygon has 9 sides, it can be named as a nonagon and since all the angles are less than 180°, it is also a convex polygon. 

Problem 2: A polygon has 20 sides. How many diagonals does it have?

Solution: According to the question,

Number of sides of a polygon (n) = 20

Now, using the formula for the number of diagonals in a polygon:

$\text{Number of diagonals}=\frac{n(n-3)}{2}$

$\text{Number of diagonals}=\frac{20(20-3)}{2}$

$\text{Number of diagonals}=10\times 17=170$

Problem 3: A traffic engineer designs a regular nonagon for a unique stop sign. What is the measure of each interior angle of this sign?

Solution: Given that the polygon is a nonagon, hence, it has 9 equal sides. 

Now, using the formula for each interior angle: 

$\text{Interior Angle}=\frac{(n-2)}{180°}n$

$\text{Interior Angle}=\frac{(9-2)\times 180°}{9}$

$\text{Interior Angle}=7\times 20°=140°$

Hence, the measure of each interior angle of this sign is 140°.