Convex Polygon

What is Convex Polygon?

We will see the definition of a convex polygon, but let us begin with a reminder of what a polygon is? It is any two-dimensional shape with straight lines and angles. So, Triangle is a polygon and so is the square. Convex polygon examples are in plenty, most of them are even not discussed regularly. Their monikers are generally based on the number of sides of the concerned 2D shape in question. Convex polygon definition is quite simple and easy to understand. A convex polygon is 2D shaped with all the interior angles less than 180-degree. A prime example of a convex polygon would be a triangle. The vertices of a convex polygon bulge away from the interior angle. It is the most important factor, which makes spotting a convex polygon definition easier.

Here, we will discuss specifically convex polygons. We will try to understand how you define convex polygon? 

We also discuss some of the properties of convex polygons that separate from the rest of the shapes? Also, how to calculate the area of a polygon? So let us jump right into it.    

Properties of Convex Polygon

There are three crucial properties of convex polygon which are mentioned below. 

  • In the case of a convex polygon, the sum of the internal angles is represented in the form of (n-2) 180*. Here, ‘n’ is the number of sides of a polygon.

  • All the interiors angles of a polygon are less than 180*

  • Concave polygon is just the opposite of convex polygon with at least one side more than 180*.

Types of Polygons

It is very important to have an answer to “what is a convex polygon”. This makes way for the type of polygons- regular and irregular.

A regular polygon is one which has all the sides of equal length, while in case of irregular polygons the length of the sides is not the same. Triangle in a convex polygon, and it has the special property of being both regular and irregular. 

Case in point, a rectangle or a square cannot be both regular and irregular polygon at the same time. 

Examples of  regular polygon- square, and equilateral triangle.

Example of irregular polygon- rectangle. And scalene triangle. 

Can a Pentagon be Both Convex and Concave?

Pentagon is any five-sided 2D shape which can be drawn on an XY plane. In case of a pentagon, the sum of interior angles can show up to 540*. 

In an ideal scenario, there are both concave and convex pentagons. Most students in the initial stages of learning form a false notion that the pentagon is rigidly but convex. 

Theorems

  • If we have a sum of all the interior angles of a regular polygon, then calculating the value of the individual interior angle is quite simple. You just need to divide the sum of all angles by the number of sides. 

  • In the case of a regular polygon with n sides, the sum of exterior angles is always equal to 360*. Thus, if we are told to find the value of an exterior angle, we just need to divide the sum of the exterior by the number of sides.  

How can you calculate the area of a convex Polygon? 

A=½  |(A1 B2 - A2 B1)+(A2 B3 - BA2)+……..+(An B1-- Bn A1)|

Using the above equation you can easily find the area of a regular convex polygon with vertices (A1, B1) , (A2, B2) ,...... (An. Bn).

Solved Examples

1. Find the Area of a Regular Polygon with Three Sides Whose Vertices are: (7, 9), (5, 2) and (-4, 5).

Here , (A1, B1)= (7,9), and 

 (A2, B2)= (5, 2), and

(A3, B3)= (-4, 5).

The formula to find the area of a convex polygon is

A=½  |(A1 B2 - A2 B1)+(A2 B3 - BA3)+……..+(A3B1-- B3 A1)|

A = ½ | (14-45) + (25+8 ) + (-36-35)|

A = ½ |-73|

A = 73/2

Therefore, the area of the convex polygon is 73/2.

2. Find the Area of a Regular Polygon with Three Sides  whose Vertices are: (10, 7), (4, 2) and (-2, 4)

Here , (A1, B1)= (10,7), and 

 (A2, B2)= (4, 2), and

(A3, B3)= (-2, 4).

The formula to find the area of a convex polygon is

A=½  |(A1 B2 - A2 B1)+(A2 B3 - BA3) + (A3B1-- B3 A1)|

A = ½ | (20-28) + (16+4 ) + (-14-40)|

A = ½ |66|

A = 66/2

Therefore, the area of the convex polygon is 73/2.

Did you know

  • Most of the irregular polygons are not actively taught and there a lot we do not know about such shapes. 

  • The term polygon has its origin from Greek word ‘poly’ meaning many and ‘gonia’ meaning angles. 

FAQ (Frequently Asked Questions)

1. Which Shapes are not Polygons?

Any shape where line segments meet erratically, and not at the vertices are considered to be as not a polygon. Henceforth, a circle is not a polygon. 


Also, the number of interior angles in a polygon is always equal to the number of sides in that polygon. Likewise, if it's a heptagon, the number of internal angles and number of sides is seven, and so on. 

2. What is a Circle if not a Polygon?

As you might have read that circles are not a polygon in the previous question. Now, this does beg a question: if circles are not a polygon, then in which category of shapes does a circle belong to? Interestingly, an hourglass is also not a polygon. It is considered to be two triangles and two polygons. 


If we have to assign a category of shape to a circle, we can say it is a polygon of infinite sides. Remember, only if we take into account infinite sides, under only, and only such circumstances can you assign the label of the polygon to a circle.