Dhristi JEE 2022-24

Define Polygons

Thinking of what polygons are? In Euclidean geometry, a polygon is a crucial concept. Polygons are closed plane figures 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 the 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

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Quadrilateral

4

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Pentagon

5

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Hexagon

6

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Heptagon

7

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Octagon

8

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Nonagon

9

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Decagon

10

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Classification of Polygons With Details 

Three-Sided Polygons or Triangles

As we know, polygons can have many sides but a three-sided polygon is known as a triangle. There are several types of triangles and we will take a brief overview of all of them:

  • Equilateral Triangle: Here, all the sides are equal and all the internal angles are 60°.

  • Isosceles Triangle: This triangle has two equal sides and the third side is of a different length. Any two of the three given internal angles are equal.

  • Scalene Triangle: all three sides and angles of the triangle are different. 


Triangles can also be described in terms of their internal angles but one should remember that all the internal angles of a triangle always add up to 180°.


A triangle with internal angles less than 90 degrees is called an acute triangle. In contrast to this, a triangle with one obtuse angle and two acute angles is called an obtuse triangle. A triangle with a right angle is known as a right-angled triangle. 


Each of these triangles can then also be classified either as equilateral, isosceles or scalene.


Four-Sided Polygons or Quadrilaterals

Four-sided polygons can be called quadrilaterals, quadrangles, or tetragons. The term quadrilateral is the most commonly used one. 


Quadrilaterals consist of various figures such as the square, the rectangle, the rhombus, and the parallelogram. The trapezium is also a quadrilateral. 


Regardless of the figure, the internal angles of all quadrilaterals add up to 360°.


  • Square: All four sides are of equal length and there are four internal right angles.

  • Rectangle: There are 4 internal right angles and the opposite sides are equal.

  • Parallelogram: The opposite sides of the figure are parallel and the opposite angles are equal.

  • Rhombus: This figure is a special type of parallelogram. Here, all four sides are of the same length.

  • Trapezium: This figure is sometimes also called a trapezoid. Two sides of this figure are parallel while the other two sides are not. The sides and the angles are all unequal.

  • Isosceles Trapezium: This is a special type of trapezium. The sides are parallel and the base angles are equal. This directly implies that non-parallel sides are also equal in length.

  • Kite: Two pairs of adjacent sides are of equal length here. The shape also has axes of symmetry present which is a distinguishing feature. 


Types of Polygon

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

  • Regular Polygon

  • Irregular Polygon

  • Concave polygon

  • Convex Polygon


Types of Polygons and its Properties

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 that has all the sides and all the angles congruent. The examples of regular polygons include plane figures such as square, rhombus, equilateral triangle, etc.


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


Concave Polygon: If one or more interior angles of a polygon measure more than 180°, 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.


Convex Polygon: If all the interior angles of a polygon strictly measure less than 180°, 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 – 2) × 180°. 


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


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


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


The triangles are formed by connecting the diagonals from one corner of a polygon = n – 2.


Angles of Polygons

As we are already familiar with, 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 − 2) × 180°.

Or

Sum of all the interior angles formula = (n − 2) π radians.


In which ‘n’ 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 angles is given as Sum of interior angles of a 4-sided polygon i.e. a quadrilateral= (4 – 2) × 180°= 2 × 180°= 360°


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 = 180°


Exterior angle = 180° – Interior angle


Area and Perimeter Formulas of Polygons

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

  • Sum of all the interior angles of an n-sided polygon = (n – 2) × 180°.

  • The angle measurement of each interior angle of an n-sided regular polygon = [(n – 2) × 180°]/n.

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

  • Number of diagonals in a polygon having n sides = n(n – 3)/2

  • Several triangles formed by connecting the diagonals from one corner of a polygon = n – 2.

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FAQs (Frequently Asked Questions)

1. How to Find the Total Number of Angle Measurements of a 5-Sided Polygon?

There is a useful formula to calculate the total (or sum) of internal angles for any polygon.(n - 2) × 180° (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 = 33 × 180 = 540°. Thus, the sum of internal angles for any pentagon (excluding complex) = 540°. These formulas come in very handy during the time of an examination. Students should keep a track of all the formulas of this chapter and make a separate formula list. 

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

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 to find each internal angle i.e 540 ÷ 5 = 108°. A regular pentagon thus contains five angles each of 108°. All this has been clearly explained meticulously in this article. Different methods are followed for calculating the angle measurements of polygons of different sides. As long as you have your concepts clear, you will be able to derive all of these easily. 

3. How do we Calculate the Area of any Regular Polygon?

Different techniques are followed for calculating the area of regular and irregular polygons. There are several classifications in place. 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) × length of the side (depicted in blue line) × 0.5 × 6 (since there are 6 triangles). 

4. How can we distinguish between convex and concave polygons?

We can easily distinguish between convex and concave polygons if we just have our basics clear. To put it simply, in any given convex polygon, the measure of the interior angle always has to be less than 180 degrees. This is why the vertices of a convex polygon always point in the outwards direction. This criterion is exactly the opposite of the criterion for the concave polygon. In a concave polygon, at least one angle has to be greater than 180 degrees. In contrast to a convex polygon, the vertices of a concave polygon can point both inwards or outwards.

5. How can students calculate the number of diagonals in any polygon?

There are dedicated formulas that help you understand the correct method of calculating the number of diagonals in any polygon. Such a polygon can be either regular or irregular. These types of questions are frequently asked as part of school and board examinations so students ought to know the formula by heart. You can refer to this article Diagonals Of A Polygon to get a better understanding of this topic. You will first be introduced to what a diagonal is and then a formula list will be provided to you. 

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