 # Dodecagon

Dodecagon is one of the types of polygons that has twelve sides. It is a regular polygon which has 12 equal sides and has 12 equal measures of angles. Irregular dodecagons have unequal sides and angles. Here, we will discuss the properties, sides, angles, area and perimeter of the twelve-sided polygon. In Geometry, we come across different shapes and figures. There are two types of shapes – geometrical and non-geometrical. Non-geometrical shapes are the ones that have no fixed 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 covered under the branch of Geometry.

## Definition and Types of Polygons

A polygon is a closed figure, formed of straight lines which are called sides. It has at least three sides and three vertices. The surface of the polygon is non-curvy or straight. There are two types of polygons – Regular and irregular. Regular polygons have equal sides as well as equal angles, whereas all the sides and angles are not equal for irregular ones. Polygons can be further classified into different types based on the number of sides as given below:

1. Three-sided Polygon→ Triangle
3. Five-sided Polygon→ Pentagon
4. Six-sided Polygon→ Hexagon
5. Seven-sided Polygon→ Septagon
6. Eight-sided Polygon→ Octagon
7. Nine-sided Polygon→ Nonagon
8. Ten-sided Polygon→ Decagon

and so on.

## Properties of a Dodecagon

• Each interior angle is equal to 150° and each exterior angle is equal to 30°.
• The total of the interior angles of a twelve-sided polygon is = (12 – 2) x 180° = 1800°.
• The total of the exterior angles of a twelve-sided polygon is 360°.
• The number of all possible diagonals in a twelve-sided polygon is given by the formula:

Total diagonals = n(n – 3)/2 = 12(12 – 3)/2 = 6 * 9 = 54

• The number of triangles formed by the diagonals from each vertex of a twelve-sided polygon is, n – 2 = 12 – 2 = 10.

### Area of Dodecagon

The total region covered inside the boundary of the Dodecagon is called the area of a Dodecagon. The area of a regular twelve-sided regular polygon of side length d is given by:

 Area = 3(2+√3)d2 ≈ 11.19615242 d2

The area calculated in terms of circumradius R of the circumscribed circle is;

 Area = 3R2

### Perimeter of Dodecagon

The perimeter is the total length of the boundaries of a twelve-sided polygon. Its formula in terms of circumradius R is given by;

 Perimeter = 12R√(2-√3) ≈ 6.2116570 R

### Examples

Q. 1: Calculate the area of the dodecagon with side length d = 10 cm.

Solution:

Number of sides = 12

Area of 12 sided polygon = 3(2+√3)d2

= 3(2+√3) x 102

= 11.19615242 x 100

Area ≈ 1119.615242 cm2

Q.2: Calculate the perimeter of a twelve-sided polygon, which is circumscribed by a radius of 5cm.

Solution: Given, radius of circumcircle = 5cm.

The formula for perimeter of dodecagon is;

P = 12R√(2-√3)

= 12 x 5 x √(2-√3)

P ≈ 31.058285 cm