Polygon

In this chapter, we will discuss Polygon. We will concentrate on the different types of the polygon as well as polygon shapes to get the right idea about it.

What are Polygons?

A closed figure which is made up of lines segments and which are not curves in two-dimensions is called a Polygon. To make a closed figure, a minimum of three line segments are required. So, in this case, a polygon having three sides is known as triangle.

polygon

What are the types of polygons?

The polygons are classified into different types depending on the sides and angles. They are:

Regular Polygon
Irregular Polygon
Convex Polygon
Concave polygon

Regular Polygon

A polygon is known as a regular polygon when all the sides and interior angles are equal.

Irregular Polygon

A polygon is an irregular polygon when all the sides and the interior angles of the polygon are of different measure.

Convex Polygon

A polygon is a convex polygon when all the interior angles of a polygon are strictly less than 180 degrees.

Concave Polygon

A polygon is known as a concave polygon when one or more interior angles of a polygon are more than 180 degrees. It can have at least four sides.

To get a clear idea, let us have a look at the table below:

Polygon	No. of Sides	No. of Diagonal	No. of vertices	Interior Angle
Triangle	3	0	3	60
Quadrilateral	4	2	4	90
Pentagon	5	5	5	108
Hexagon	6	9	6	120
Heptagon	7	14	7	128.571
Octagon	8	20	8	135
Nonagon	9	27	9	140
Decagon	10	35	10	144
Hendecagon	11	44	11	147.273
Dodecagon	12	54	12	150
Triskaidecagon	13	65	13	158.308
Tetrakaidecagon	14	77	14	154.286
Pentadecagon	15	90	15	156

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Three Dimensional shapes of Polygon

Let us now look at the three-dimensional shapes of Polygon (3D shape)

three-dimensional shapes of Polygon (3D shape)

Students should work on different problems on the polygon to get the best idea about it. Here, we will work on the solved problems to provide the right understanding about it.

Problem 1

In a circle of radius 10 cm, a 6 sided regular polygon (hexagon) is inscribed. Can you find the length of one side of the hexagon?

Solution: Here, Angle AOB is given by

angle (AOB) = 360^o / 6 = 60^o

Here, OA = OB = 10 cm, so triangle OAB is isosceles for which it gives

angle (OAB) = angle (OBA)

Therefore all three angles of the triangle are equal, and it is an equilateral triangle. Hence

AB = OA = OB = 10 cm.

Problem 2

In a 5 sided regular polygon (pentagon), a circle of radius 6 cm is inscribed. Can you find the length of one side of the pentagon? (approximate your answer to two decimal places).

Solution

Let us assume ‘t’ to be the size of angle AOB,

So, t = 360^o / 5 = 72^o

The polygon is regular and so OA = OB.

Now assume M to be the midpoint of AB so that OM is perpendicular to AB. The radius of the inscribed circle is OM which is equal to 6 cm. Now, according to the right angle
tan(t / 2) = MB / OM

Here, the side of the pentagon is twice MB, therefore,

the side of pentagon = 2 OM tan(t / 2) = 8.7 cm (answer rounded to two decimal places)

Problem 3: An exterior angle of a regular polygon measures 36°. How many sides does the polygon have?

Solution

In order to find the measure of each interior angle of a regular polygon, we need to use the formula

360/n, where n is the number of sides.

By applying the formula to solve for n, we get

360/n = 36

360 = 36n

360/36 = n

n = 10

Problem 4: The sum of the interior angles of a polygon is 180° How many sides does it have?

Solution: In order to find the sum of the interior angles of a polygon, we need to use the formula:
180 (n - 2) where n is the number of sides.

Now, by applying the formula to solve for n, we get

180 (n - 2)   =   180
n - 2   =   1
n   =   3

Important Chapters:

Polygon

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Polygon

What are Polygons?

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