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# Dodecagon

In geometry, one studies about different shapes and figures. There are two types of shapes - geometrical and non geometrical. Non geometrical shapes are the ones that has no fixed and shape or angles. On the other hand, geometrical shapes are the shapes that have a definite form. Some are made up of angles and straight lines, while some are composed of curves and arcs. All geometrical shapes are studied under the branch of geometry.

Polygon is an important class of figures. A polygon is defined as a closed figure that is composed of straight lines known as sides. It does have at least 3 sides. There are no curvy surfaces in a polygon. There are mainly two types of polygons - Regular and irregular. Regular polygons do have equal angles as well as equal sides. On the other hand, irregular ones do not possess all sides and angles equal.

Polygons can also be classified in different types on the basis of number of side as explained below -

1) Three-sided Polygon - Triangle

3) Five-sided Polygon - Pentagon
4) Six-sided Polygon - Hexagon
5) Seven-sided Polygon - Septagon
4) Eight-sided Polygon - Octagon
and so on.

Dodecagon is one of the types of polygon which has twelve sides. This is one of the regular polygons and is named dodecagon since it has 12 sides and has 12 angles. There are both regular and irregular dodecagons. Here, we will discuss more about the properties of the regular 12 sided polygon.

## Properties of a Dodecagon

Given below are some facts about dodecagon:
• The total interior angle of a 12 sided polygon is = (12 - 2) 180 degrees = 1800 degrees.
• The internal angle at each vertex of an regular dodecagon is equal to = $\frac{1800}{12}$ = 150 degrees. While the exterior angle at a single vertex is = $\frac{360}{12}$ = 30 degrees.
• The total exterior angle of a 12 sided polygon is 360 degrees.
• The number of all possible diagonals in a 12 sided polygon is given by the formula, Total diagonals = $\frac{n(n - 3)}{2}$ = $\frac{12(12 - 3)}{2}$ = 6 * 9 = 54.
• The number of triangles formed by the diagonals from each vertex of a 12 sided polygon is, n - 2 = 12 - 2 = 10.

## Dodecagon Sides

A dodecagon is a polygon with exactly 12 sides.

Dodecagon Shape:

A regular dodecagon is shown below:

## Dodecagon Angles

For Regular Dodecagon

Interior Angle
Each interior angle is equal to 150o.
Let us find sum of interior angles for a Dodecagon:
Sum of interior angles of any polygon with n sides = (n - 2)180o .

For a Dodecagon, n = 12.

So, sum of angles of a Dodecagon = (12 - 2) 180o

= 10 * 180o

= 1800o.

=> Sum of interior angles of a dodecagon = 1800o.

Exterior Angle

Exterior angle forms a linear pair with the interior angle.

ie. each exterior angle = 180o - 150o = 30o.

=> Exterior angle of a regular dodecagon = 30o.

For Irregular Dodecagon

In case of an irregular dodecagon, interior angles are based on the lengths of sides.

## Area of a Dodecagon

The total space inside the boundary of the Dodecagon is called as the area of a Dodecagon. The area of an regular Dodecagon of side length d is given by,

Area = 3(2 + $\sqrt{3}$)d$^2$

The area calculated using the radius R of the circumscribed circle is,

Area = 3R2.

### How to Find the Area of a Dodecagon?

Given below are some of the examples on regular dodecagon.

### Solved Examples

Question 1: Calculate the area of the dodecagon with side length d = 5 cm.
Solution:

Dodecagon sides = 12

Area of 12 sided polygon =  3(2 + $\sqrt{3}$)d$^2$

=  3(2 + $\sqrt{3}$)5$^2$

= 3(2 + $\sqrt{3}$) * 25

= 75(2 + $\sqrt{3}$)

= 279.90 cm2.

Question 2: Find the area of a 12 sided polygon with a circumscribed circle radius of 4 cm.
Solution:

Area of the 12 sided polygon in a circumscribed circle of radius R is,

Area = 3R2.

Area = 3 * 4

= 3 * 16

= 48 cm2.

## Practice Problems on 12 Sided Polygon

Given below are some of the practice problems on regular 12 sided polygon.

### Practice Problems

Question 1: Calculate the area of the dodecagon with a side length of 8 cm.
Question 2: Calculate the area of the dodecagon with a circumscribed circle radius of 6.5 cm.
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