Polygon: Definition, Shape, Sides, Types, Examples & Formula

Polygon: A Polygon is a two-dimensional geometric figure with a fixed number of sides. It is commonly seen in our day-to-day activities such as square, rhombus, rectangle, triangle, and much more. In other words, polygons are simple closed shapes formed by only straight line segments. The minimum number of line segments required to construct a polygon is three. A regular polygon has equal sides and angles.

Circles and other three-dimensional figures such as spheres, cylinders, and cones, as well as any shapes with curved surfaces and open shapes, are not polygons. In this article, let’s discuss polygon and its properties in detail.

Table of Contents – Polygon

1. Introduction to Polygons
2. What is a Curve?
3. What are the Types of Curves?
4. What is Polygon?
5. What is Polygon Shape?
6. Types of Polygon
7. Angles of Polygon
8. Polygon Formula
9. Solved Examples on Polygon
10. Summary
11. Polygon FAQs

Introduction to Polygons

We have seen many plane surfaces in our daily lives, such as the page of a notebook, the blackboard in our classroom, the top surface of our study table, a drawing board, and so on.

They are the perfect examples of a plane surface.

Now look at these figures \((i), (ii), (iii), (iv), (v)\) and \(\left( {iv} \right).\)

Are they closed figures? What is the difference between them?

If your answer is Yes, then you got it right. All the above figures are closed curves. \((i), (ii), (iii), (iv)\) and \((vi\)) are formed by joining the straight line segments. But \((v)\) is formed by a line segment and a curved line.

So, \((i), (ii), (iii)\), and \((iv)\) are simple closed curves. They are also called polygons. So, a shape is a polygon if it is a simple closed figure made up entirely of straight-line segments.

Learn Properties of Polygon Here

What is a Curve?

A figure that can be drawn by moving a point in a two-dimensional plane without any sharp bends is known as a curve.

What are the Types of Curves?

(i) Open Curve – A curve that does not cut itself is called an open curve.

(ii) Closed Curve – A curve that cuts itself at least at one point is called a closed curve.

(iii) Simple Closed Curve – A closed curve is called a simple closed curve if it does pass through one point more than once.

What is Polygon?

The word is derived from the Greek word poly means many and gon means angle. The definition of a polygon is – a simple closed curve formed by three or more line segments such that

• (i) no two-line segments intersect except at their endpoints.
• (ii) no two line segments with a common endpoint are coincident.

In other words, a polygon is a simple closed two-dimensional shape formed by joining the straight line segments.

Examples: equilateral triangle, square, scalene triangle, rectangle, etc.

Note: The circle is a simple closed curve, but it is not a polygon as it is not made by line segments.

What is Polygon Shape?

The shape of the polygon depends upon the number of sides. For example, if the number of sides of a polygon is three then it is a triangle, if the number of sides of a polygon is four then it is a quadrilateral, if the number of sides of a polygon is five then it is a pentagon and so on.

Let’s have a look at the shapes given below.

What are the Types of Polygon?

There are three major types of polygons:

1. Regular or Irregular
2. Concave or Convex
3. Simple or Complex

(i) Regular or Irregular Polygon – If all sides and all interior angles of a polygon are equal, then it is called a regular polygon.

Examples: equilateral triangle, square, etc.

If all sides and all interior angles of a polygon are not equal, then it is called an irregular polygon.

Examples: scalene triangle, rectangle, etc.

(ii) Concave or Convex Polygon – A polygon in which at least one of the interior angles is more than a straight angle (or \({180^ \circ }\)) is called a concave polygon.

A polygon in which each interior angle is less than a straight angle (or \({180^ \circ }\)) is called a convex polygon.

(iii) Simple or Complex Polygon – A simple polygon is a polygon that is made up of only non-intersecting straight line segments.

A polygon whose sides cross over each other at more than one point is called a complex polygon.

What are the Angles of a Polygon?

A polygon has two types of angles:

(i) Interior angles – Angles that are formed at the vertices, inside the polygon.

(ii) Exterior angles – Angles that are formed outside the polygon when a side of a polygon is extended. It is adjacent to (beside) the interior angle.

What are the Polygon Formulas?

The important formulas related to a polygon are:

1. To calculate the sum of interior angles of a polygon, first, divide the polygon into triangles and then multiply the number of triangles in the polygon by \({180^{\rm{o}}}.\)
2. The formula for calculating the sum of interior angles is \(\left({n – 2} \right) \times 180^\circ \) where \(n\) is the number of sides.
3. All the interior angles of a regular polygon are equal. The formula for calculating the measure of an interior angle of a polygon is given by:
   \({\rm{Interior}}\,{\rm{angle}}\,{\rm{of}}\,{\rm{a}}\,{\rm{polygon}} = \frac{{{\rm{Sum}}\,{\rm{of}}\,{\rm{interior}}\,{\rm{angles}}}}{{{\rm{Number}}\,{\rm{of}}\,{\rm{sides}}}}\)
4. The sum of all the exterior angles of a polygon is \({360^{\rm{o}}}.\)
5. The formula for calculating the measure of an exterior angle is given by
   \({\rm{Exterior}}\,{\rm{angle}}\,{\rm{of}}\,{\rm{a}}\,{\rm{polygon}} = \frac{{{{360}^{\rm{o}}}}}{{{\rm{Number}}\,{\rm{of}}\,{\rm{sides}}}}\)

Solved Examples – Polygon

Q.1. Examine whether the following are polygons, if not why?

Ans: The shape of a polygon is a simple closed curve made up of straight-line segments such that no two line segments intersect each other at more than one point and no two line segments are overlapping.
In the given figure \((i)\) the curve is not closed. So, it is not a polygon.
In the given figure \((ii)\) the curve is closed and made up of four straight line segments. So, it is a polygon.
In the given figure \((iii)\) the curve is closed but it is not made up of straight-line segments. So, it is not a polygon.
Therefore, only \((ii)\) is a polygon.

Q.2. Count the number of sides of the polygons given below and name them:

Ans: (i) In this figure the number of sides of the polygon is five, so it is a pentagon,
(ii) In this figure the number of sides of the polygon is eight, so it is an octagon.
(iii) In this figure the number of sides of the polygon is six, so it is a hexagon.
(iv)In this figure the number of sides of the polygon is three, so it is a triangle or a trigon.

Q.3. Identify the regular polygons among the figures given below:

Ans: We know that a regular polygon has all the sides and angles equal.
As we can see that the figure \(\left( i \right),\,\left( {iv} \right)\) and \(\left( {vi} \right)\) have all the sides and angles equal.
So, \(\left( i \right),\,\left( {iv} \right)\) and \(\left( {vi} \right)\) are the regular polygons.

Q.4. The sum of the interior angles of a polygon is \({1620^{\rm{o}}}.\) Find the number of sides of the polygon.
Ans: Given, the sum of the interior angles of polygon \({1620^{\rm{o}}}.\)
Let the number of sides of the polygon \( = n\)
We know that, the sum of all the interior angles of a polygon \( = n – 2 \times {180^{\rm{o}}} \Rightarrow {1620^{\rm{o}}} = n – 2 \times {180^{\rm{o}}} \Rightarrow n = 11\)

Q.5. If one angle of a five-sided polygon is \(140^\circ \) and the remaining angles are in the ratio \(1:2:3:4.\) Find the measure of the greatest angle.
Ans:  We know that the sum of all the interior angles of a polygon \( = n – 2 \times {180^{\rm{o}}} \Rightarrow \left( {5 – 2} \right) \times {180^{\rm{o}}} = {540^{\rm{o}}}\)
Let the other angles are \(x,\,2x,\,3x,\) and \(4x\) \( \Rightarrow 140^\circ  + x + 2x + 3x + 4x = 540^\circ \)
\( \Rightarrow 10x = 400 \Rightarrow x = 40\)
Hence, the greatest angle \( = 4x = 4 \times 40 = 160^\circ \)

Summary

In this article, we have learned about the basic definition of a curve and we have discussed the definition of a polygon, and its types, shapes, and other different properties.  By going through the article, you would have also got the fair idea to differentiate between a regular or irregular polygon, concave or convex polygon, simple or complex polygon.

Learn Properties of Polygon Here

Frequently Asked Questions (FAQ) – Polygon

Q.1. What is a \(12\) sided polygon called?
Ans: The polygon with \(12\) sides is called Dodecagon.

Q.2. How do you identify a polygon?
Ans: A polygon is made of line segments and is a closed \(2 – D\) figure.

Q.3. How many sides are in a polygon?
Ans: Since the minimum number of sides of a polygon is three, the polygon can have three or more than three sides.

Q.4. Is circle a regular polygon?
Ans: The shape of a polygon is a simple closed curve made up of straight-line segments such that no two line segments intersect each other at more than one point and no two line segments are overlapping. A circle is a simple closed figure, but it is not made up of straight-line segments. Therefore, the circle is not a polygon. 

Q.5. Is a rhombus a polygon?
Ans: Yes, a rhombus is made up of four straight-line segments. It is a type of quadrilateral.

Q.6. What is a polygon shape?
Ans: The shape of a polygon is a \(2-D\) closed shape made up of straight-line segments.

Q.7. Can a polygon be open?
Ans: A polygon cannot be open since it is a simple closed curve.

Check Examples on Area of Polygon Here

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We hope this article has helped you in understanding the concept of Polygon. However, you have any doubts or queries related to this article, feel to drop your comments in the comment box below and we will be happy to help you.