Classification of Polygon: Do you know what a polygon is? Polygons, after all, are all around us! Polygons make up the majority of the shapes you see or study daily. A polygon is what you perceive when you view a rectangular wall. A polygon is the front view of a dice with a square form. A polygon is formed by a pizza slice, which is triangular. A polygon is there on the screen of your television or laptop. Let us understand in detail what a polygon is.

This article will discuss the polygon definition and its classification based on the number of sides and the solved examples.

What is a Polygon?

A polygon is a simple closed curve made up of straight-line segments only.

The adjoining figure is a polygon since it is
– closed
– bounded by five line segments \(AB,BC,CD,DE\) and \(AE.\)

Also, it is clear from the given polygon that:
(i) the five-line segments \(AB,BC,CD,DE\) and \(AE\) intersect at their endpoints.
(ii) two line segments with a common vertex are not collinear, i.e. the angle at any vertex is not \({180^ \circ }.\)

Learn All the Concepts on Polygons

Parts of a Polygon

(i) Line segments forming a polygon are called sides of the polygon.

(ii) The point where two sides of a polygon meet are called the polygon’s vertex.

(iii) The line segment containing two non-adjacent vertices is called the diagonal of the polygon.

(iv) The angle formed at the vertices inside the closed figure are called interior angles.

(v) Angles formed outside the polygon when a side of a polygon is extended are called the exterior angles of a polygon. It is adjacent to (beside) the interior angle.

Note: The number of sides in a polygon is equal to the number of vertices.

Classification of Polygons

Polygons are classified according to the number of sides or vertices they have. 

1. Triangle: A triangle is a polygon with three sides. It is the smallest polygon. They are further classified into different categories, such as:

(i) Scalene Triangle: A triangle with all three sides different in length is called a scalene triangle.

(ii) Isosceles Triangle: A triangle in which two sides are of equal lengths is called an isosceles triangle.

(iii) Equilateral Triangle: A triangle with all three sides equal is called an equilateral triangle. And, all angles of an equilateral triangle measures \({60^ \circ }.\)

The sum of the interior angle of a triangle is \({180^ \circ }.\)

2. Quadrilateral: The quadrilateral is a four-sided polygon or a quadrangle.

The different types of quadrilaterals are square, rectangle, rhombus, parallelogram, trapezium and kite.

The sum of the interior angle of a quadrilateral is \({360^ \circ }.\)

3. Pentagon: Pentagon is a five-sided polygon. A pentagon is a figure obtained by joining the points of five-line segments in the same plane.

A regular pentagon has all five sides of the polygon equal in length. If the length of the sides is not equal, then it is called an irregular pentagon.
The sum of the interior angle of a pentagon is \({540^ \circ }.\)

4. Hexagon: Hexagon is a polygon that has \(6\) sides and \(6\) vertices. A regular hexagon has all six sides equal in length. And, its interior angles and exterior angles are also equal in measure.

The sum of the interior angle of a hexagon is \({720^ \circ }.\)

5. Heptagon: Heptagon is a polygon that has seven sides and seven vertices. A regular heptagon has all seven sides equal in length. And, its interior angles and exterior angles are also equal in measure.

The sum of the interior angle of a heptagon is \({900^ \circ }.\)

6. Octagon: Octagon is a polygon that has eight sides and eight vertices. A regular octagon has all eight sides equal in length. And, its interior angles and exterior angles are also equal in measure.

The sum of the interior angle of an octagon is \({1080^ \circ }.\)

7. Nonagon: Nonagon is a polygon that has nine sides and nine vertices. A regular nonagon has all nine sides equal in length. And, its interior angles and exterior angles are also equal in measure.

The sum of the interior angle of the nonagon is \({1260^ \circ }.\)

8. Decagon: Decagon is a polygon that has ten sides and ten vertices. A regular decagon has all ten sides equal in length. And, its interior angles and exterior angles are also equal in measure.

The sum of the interior angle of a decagon is \({1440^ \circ }.\)
Classification of polygons having the number of sides \(11\) to \(20\):

Name of the polygon	Number of sides	Number of vertices
Hendecagon	\(11\)	\(11\)
Dodecagon	\(12\)	\(12\)
Trislaidecagon	\(13\)	\(13\)
Tetrakaidecagon	\(14\)	\(14\)
Petadecagon	\(15\)	\(15\)
Hexakaidecagon	\(16\)	\(16\)
Heptadecagon	\(17\)	\(17\)
Octakaidecagon	\(18\)	\(18\)
Enneadecagon	\(19\)	\(19\)
Icosagon	\(20\)	\(20\)

Polygon Formulae

The important formulae related to the 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 \({180^ \circ }.\)
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:
\({\text{Interior}}\,{\text{angle}}\,{\text{of}}\,{\text{a}}\,{\text{polygon}} = \frac{{{\text{Sum}}\,{\text{of}}\,{\text{interior}}\,{\text{angles}}}}{{{\text{Number}}\,{\text{of}}\,{\text{sides}}}}\)
4. The sum of all the exterior angles of a polygon is \({360^ \circ }.\)
5. The formula for calculating the measure of an exterior angle of a regular polygon is given by \({\text{Exterior}}\,{\text{angle}}\,{\text{of}}\,{\text{a}}\,{\text{polygon}} = \frac{{360}}{{{\text{Number}}\,{\text{of}}\,{\text{sides}}}}\)
6. For an \(n\)-sided regular polygon, the number of diagonals \( = \frac{{n\left({n – 3} \right)}}{2},\) where \(n\) is the number of sides of the polygon.

Solved Examples – Classification of Polygons

Q.1. A polygon has 27 diagonals. Identify the number of sides present in the polygon.
Ans: We know that,
Number of diagonals in a polygon \( = \frac{{n\left({n – 3} \right)}}{2}\)
Where \(n\) is the number of sides of the polygon.
Now, according to the given question,
\(\frac{{n\left({n – 3} \right)}}{2} = 27n\left({n – 3} \right) = 54{n^2} – 3n – 54 = 0{n^2} – 9n + 6n – 54\)
\(=0n\left({n – 9} \right) – 6\left({n – 9} \right) = 0\left({n + 6} \right)\left({n – 9} \right) = 0n = 9, – 6\)
Since \(n\) is a negative value, the number of sides present in the polygon is \(9.\)

Q.2. Classify the polygons based on the number of sides

Answer: (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. What is the measure of all the angles in a square?
Ans: We know square is a regular polygon with each angle measures \({90^ \circ }.\)
Therefore, the sum of four angles in a square is:
\({90^ \circ } +{90^ \circ } + {90^ \circ } + {90^ \circ } = {360^ \circ }\)
Therefore, the sum of the measure of all the angles of a square is \({360^ \circ }.\)
Also, the square is a quadrilateral, and the sum of interior angles of any quadrilateral is \({360^ \circ }.\)

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

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 \(= \left({n – 2} \right) \times{180^ \circ }\)
\( \Rightarrow \left({5 – 2} \right) \times {180^ \circ } = {540^ \circ }\)
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 discussed the definition of a polygon and its different parts. We also came across the classification of polygons based on the number of sides. A three-sided polygon is called a triangle; a four-sided polygon is a quadrilateral; a five-sided polygon is a pentagon; a six-sided polygon is a hexagon, etc. One can easily classify the polygon studied in this article by looking at the number of sides.

Frequently Asked Questions (FAQs)

Q.1. What are polygons and the classification of the polygon?     
Ans: A polygon is a simple closed curve made up of straight line segments only. And based on the number of sides, polygons are classified as follows:

Name of the Polygon	Number of sides	Number of vertices
Triangle (Trigon)	\(3\)	\(3\)
Quadrilateral (four-gon)	\(4\)	\(4\)
Pentagon	\(5\)	\(5\)
Hexagon	\(6\)	\(6\)
Heptagon	\(7\)	\(7\)
Octagon	\(8\)	\(8\)
Nonagon	\(9\)	\(9\)
Decagon	\(10\)	\(10\)
Hendecagon	\(11\)	\(11\)
Dodecagon	\(12\)	\(12\)
Triskaidecagon	\(13\)	\(13\)
Tetrakaidecagon	\(14\)	\(14\)
Petadecagon	\(15\)	\(15\)
Hexakaidecagon	\(16\)	\(16\)
Heptadecagon	\(17\)	\(17\)
Octakaidecagon	\(18\)	\(18\)
Enneadecagon	\(19\)	\(19\)
Icosagon	\(20\)	\(20\)

Q.2. What is the classification for a 3-sided polygon?
Ans: The \(3\)-sided polygon is called a triangle or a trigon.

Q.3. What are the 3 characteristics of a polygon?
Ans: The three characteristics of a polygon are:
1. The number of sides of the shape.
2. The angles between the sides of the shape
3. The length of the sides of the shape

Q.4. How do you identify a polygon?
Ans: A polygon can be identified by its sides. A polygon is made up of all line segments. The minimum number of sides required to draw a polygon is three.

Q.5. Is a circle a regular polygon?
Ans: A circle is a simple closed figure, but it is not made up of straight-line segments. Therefore, the circle is not a polygon.