Polygon Formula

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What is Polygon?

A polygon is a simple closed curve. A two-dimensional closed figure bounded with three or more than three straight lines is called a polygon. Triangles, square, rectangle, pentagon, hexagon, are some examples of polygons.

The segments are referred to as the sides of the polygon. The points at which the segments meet are called vertices. Segments that share a vertex are called adjacent sides. A segment whose endpoints are nonadjacent vertices is called a diagonal. See the picture below.

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The below figures show some of the examples of polygons or polygonal curves( a closed curve that is not a polygon).

(Image to be added soon)(Image to be added soon)(Image to be added soon)                       

In figure you can see that all the shapes are polygons, as all the shapes are drawn joining the straight lines only. There are no curved lines..

But in figure the shape is not a polygon because it is not fully connected and also has curved lines and such shapes that have curves are called a closed curve that is not a polygon.

In this article let us study what is polygon and regular polygon formulas.


Types of Polygons

As we know what is the meaning of polygon let us understand different types of polygons. They are as follows,

  • Regular Polygon

  • Irregular Polygon

  • Convex Polygon

  • Concave polygon


Regular Polygon

Polygons whose sides and angles are of equal lengths are called regular polygons.

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Irregular Polygon

Polygons with different sizes and different interior angles are called irregular polygons.

(Image to be added soon)


Convex Polygon

Polygons with interior angles less than 1800 are called convex polygons.

(Image to be added soon)


Concave Polygons

Polygons with interior angles greater than 1800 are called concave polygons.


Some Popular Polygons

Here is the list of some of the regular polygons with the number of polygon sides, shapes, and measures of its interior angles.

Name 

Number of Polygon Sides

Shape 

Measures of interior angles

Triangle

3

(Image to be added soon)

600

Quadrilateral

4

(Image to be added soon)

900

Pentagon

5

(Image to be added soon)

1080

Hexagon

6

(Image to be added soon)

1200

Heptagon

7

(Image to be added soon)

128.5710

Octagon

8

(Image to be added soon)

1350

Nonagon 

9

(Image to be added soon)

1400

Decagon

10

(Image to be added soon)

1440

Hendecagon

11

(Image to be added soon)

147.2730

Dodecagon

12

(Image to be added soon)

1500

Triskaidecagon

13

-

152.3080

Tetrakaidecagon

14

-

154.2860

Pentadecagon

15

-

1560

n-gon

n

(Image to be added soon)

(n−2) x 180° / n


Regular Polygon Formulas 

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Some of the regular polygon formulas related to polygons are:

  • Interior Angle: An angle inside the polygon at one of its vertices is called the interior angle. Angle Q is an interior angle of quadrilateral.

  • Exterior Angle: An angle outside the polygon formed by one of its sides and the extension of an adjacent side is called the exterior angle of the quadrilateral.in the above quadrilateral angle ADZ, YUA, YUX, and QUX are the exterior angles.

  • The sum of the measures of all the exterior angles of a polygon is 360°.

  • Interior angle + corresponding exterior angle = 1800

  • The sum of the measures of the interior angles of a polygon with n sides is given by the general formula  (n–2)180.

Sum of Interior angles of Polygon(IA) = (n-2) x 180

  • The measure of an exterior angle of a regular n - sided polygon is given by the formula 360/n

Exterior angle of a regular polygon(EA) = 360/n

  • The measure of an interior angle of a regular n- sided polygon is \[\frac{(n-2)180}{n}\]

Interior angle of a regular polygon = \[\frac{(n-2)180}{n}\]

  • The number of diagonal of n- sided polygon is \[\frac{n(n-3)}{2}\]

Diagonal of Polygon = \[\frac{n(n-3)}{2}\]

  • Perimeter P of a regular n-sided Polygon with s as the length of the sides is given by n x s  

Perimeter of Polygon(P) = n x s

  • Area of polygon formula of a regular n-sided polygon with s as the length of the sides is given by s/2tan(180/n)

Area of Polygon(A) = s/ 2 tan (180/n)


Solved Examples

Example 1:  A polygon is an octagon and its side length is 6 cm. Calculate its perimeter and value of one interior angle.

Solution:

The polygon is an octagon, so we have, n = 8

Length of one side, s = 6 cm

The perimeter of the octagon

P = n × s

P = 8 × 6

   =48 cm.

Now, for the interior angle, we have

Interior angle of a regular polygon = \[\frac{(n-2)180}{n}\]

= ( 8 - 2) x 180/ 8

= 6 x 180 /8

= 1350

Therefore the perimeter of the octagon is 48cm and the value of one of the interior angles is 1350.


Example 2: Calculate the measure of one interior angle of a regular hexadecagon (16 sided polygon)?

Solution:

The polygon is an hexadecagon, so we have, n = 16

Interior angle of a regular polygon (IA) = \[\frac{(n-2)180}{n}\]

IA = ( 16 - 2) x 180 / 16

= 14 x 180 /16

= 157.50

Therefore measure of interior angle of a regular hexagon is 157.50.


Example 3: Calculate the measure of 1 exterior angle of a regular pentagon?

Solution: the polygon is a pentagon , so we have n = 5

An exterior angle of a regular polygon (EA) = 360/n

= 360/ 5

= 720

Therefore the measure of an exterior angle of a regular pentagon is 720.


Quiz Time

  1. What is the measure of 1 exterior angle of a regular decagon (10 sided polygon)?

  2. If each exterior angle measures 10°, how many sides does this polygon have?

FAQ (Frequently Asked Questions)

1. How to Calculate the Internal Angle of a Polygon?

You can calculate the internal angles of a polygon using one of the methods mentioned below:

Divide the polygon into triangles.

We can split the polygon into triangles. Once you have done that you can use the angle sum of a triangle to find out the sum of all of the angles. If the shape is regular you can calculate the measure of each angle by dividing the sum of all angles by the number of sides of the shape.

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In the example above we just divide 180 by 3 (as we can fit 3 triangles into the pentagon). This also gives us 540⁰. You then divide this by 5 to give us 108⁰ this is therefore the size of each internal angle. 

2. What is a One-Sided Polygon Called?

There is no polygon with one side because the properties of polygons itself state that a two-dimensional closed figure bounded with three or more than three straight lines is called a polygon.

So the minimum number of slides to form a polygon is three. And it should be a closed figure. 

The line is an example of a dimensional shape. It has only one dimension and it is an open figure. So the line is not called a polygon. 

A circle can also be not considered as a one-sided polygon because it is not made of line segments, it is made up of curved lines.