Dodecagon

What’s a Dodecagon?

In geometry, we come across many shapes. These are mostly 2-dimensional or 3 dimensional in nature. A closed figure which has at least three sides is called a polygon. There can be many different kinds of polygons. You have already heard of a triangle (has 3 sides) and rectangle (has 4 sides). Similarly, a dodecagon is also a polygon. It has 12 sides. Like other polygons, a dodecagon can be either regular or irregular. In this section, we shall be discussing all the different properties of dodecagon as well as interesting facts about them.

Polygons and their Different Types

Any closed figure constructed out of straight lines is termed as a polygon. These straight lines are known as the sides of the polygon. In order to create a polygon, we need at least three different sides as well as vertices. Polygons can be broadly divided into two types:

  • Regular Polygon: It consists of equal sides as well as equal angles.

  • Irregular Polygon: Here, the sides and angles vary.

Further Classification of Polygons According to their number of sides: 

Three-sided Polygon

Triangle

Four-sided Polygon

Quadrilateral

Five-sided Polygon

Pentagon

Six-sided Polygon

Hexagon

Seven-sided Polygon

Septagon

Eight-sided Polygon

Octagon

Nine-sided Polygon

Nonagon

Ten-sided Polygon

Decagon

What Is An Irregular Dodecagon?

As per the properties of irregular polygons, irregular dodecagons have unequal sides or unequal angles. Both of these fall under the dodecagon category. A very good example of an irregular dodecagon is the Red Cross symbol. The Red Cross is a dodecagon which has equal sides but 8 interior angles measure 90 degrees while 4 of these angles measure 270 degrees.

Unique Properties of a Dodecagon

Dodecagon is one of the more fascinating geometrical figures. Down below is a list of some of the unique properties of dodecagons:

  • One interior angle of a regular dodecagon is 150° which sums up to a total of 1800°.

  • The sum of all the exterior angles of a polygon is always 360°, i.e., so for dodecagons, each angle is of 30°.

  • The total number of diagonals of a dodecagon is 54. It can be found from the formula given below:

 Total no. of diagonals = n(n-3)/2 { n= number of side}

                             = 12(12-3)/2

                           = 6*9 = 54

  • The total number of triangles in a dodecagon that can be formed is 10; it is simply given by n-2, where n is the number of sides. 

The Perimeter of a Dodecagon

A perimeter can be best described as the total length of the boundary of a polygon. So, as for dodecagon, it can be easily given by 12S {where s the length of one side}.

 But, the perimeter of a dodecagon can also be calculated in terms of its circumradius R. It is given by:

 Perimeter = 12R √(2-√3) ≈ 6.2116570 R

How to find the area of a Dodecagon Calculator

It is imperative to understand that the area of a dodecagon can also be found in one of the two ways:

1) Area of the dodecagon using the length of the side as d:

Area = 3(2+√3) d2 ≈ 11.19615242 d2

2) Area of the dodecagon using the circum-radius R

Area = 3R2

Solved Examples on dodecagons

  1. Calculate the perimeter of the dodecagon whose circumradius is 10 cm.

Solution:

R = 10cm

Using the formulae of the perimeter for dodecagon,

P = 12R√(2-√3)

P = 12 x 10 x √(2-√3)

P = 62.11657 cm

  1. What would be the area of a dodecagon whose length of the side is 12 cm?

Solution:

Length of the side: 12 cm

So, the formula of area of a dodecagon is

A = 3(2+√3) x 122

A = 11.19615242 x 144

A = 1612.2456 cm2

FAQ (Frequently Asked Questions)

1. What is the difference between dodecagon and dodecahedron?

The 12-sided polygon or commonly known as the dodecagon is a 2D geometrical figure. On the contrary, a Dodecahedron is a 12-faced polyhedron. While a dodecagon has 12 faces and 12 vertices, the dodecahedron consists of 12 faces with each face consisting of 5 edges along with 20 vertices. One of the main differences is that dodecahedron is a 3D shape and so, unlike the 2-dimensional figure, it also has volume. As a result, you can calculate the volume of a dodecahedron which you will not be able to for a dodecagon since it is just a 2-d figure.

2. Can the formula of triangles be used to find the area of a dodecagon?

Yes, it is possible to find the area of a triangle and therefore, find the area of a dodecagon. For this method, you have to consider the regular dodecagon to have 12 isosceles triangles with one angle at 30 degrees and the remaining two angles at 75 degrees. On that basis, if you have the sides of the dodecagon triangle, you can take one as the base, the other as the height (you may have to use Pythagoras theorem) and calculate the area of one triangle. Area of triangle = ½ * base * height. Now multiply it by 12 and you get the area of the dodecagon.