Dodecagon
Dodecagon is one of the types of polygons that has 12 sides, 12 vertices and 12 angles. Similar to other polygons, a dodecagon is also a two-dimensional plane figure. A regular dodecagon polygon has 12 equal sides and has 12 equal measures of angles. Irregular dodecagons have unequal sides and angles. Also, a dodecagon can be a convex polygon or concave polygon. The sum of all the interior angles of the dodecagon is equal to 1800°.
The word dodecagon is derived from Greek words, ‘dodeka‘ which means ‘twelve’ and ‘gon‘ means ‘sides’. Here, we will discuss the properties, sides, angles, area and perimeter of the twelve-sided polygon (dodecagon).
Table of contents: |
What is a Dodecagon?
A dodecagon is a closed figure that has 12 sides and 12 angles. It has 12 vertices, each of which is connected to two sides. See the below figure which is an example of a dodecagon.
Depending on the sides and angles a dodecagon can be a regular or irregular polygon and a convex or concave polygon. Let us now discuss the types of a dodecagon.
Types of Dodecagon
There are basically four types of a dodecagon, they are:
- Regular dodecagon
- Irregular dodecagon
- Convex dodecagon
- Concave dodecagon
Let us discuss them all one by one.
Regular dodecagon
A regular dodecagon has all its 12 sides equal in length and all the angles have equal measures. All the 12 vertices are equidistant from the center of the dodecagon. For a regular dodecagon each of its interior angle measure 150°.
A regular dodecagon is a symmetrical polygon.
Irregular dodecagon
An irregular dodecagon does not have all its 12 sides equal in length. Also, the measure of 12 angles of an irregular dodecagon, may vary from each other. It is an unsymmetrical polygon.
Convex dodecagon
A usual dodecagon that has all its vertices pointed outside the center is a convex polygon. No line segments between the vertices go inside the convex dodecagon. All the interior angles will be less than 180° here.
Concave dodecagon
If any one of the interior angles of a dodecagon is greater than 180 degrees, then it is called a concave dodecagon. We may also say if any of the vertices are pointed towards the center or inwards, then it is a concave dodecagon.
Properties of a Dodecagon
- A dodecagon has 12 sides, 12 vertices and 12 angles
- Each interior angle is equal to 150° and each exterior angle is equal to 30°.
- Interior angles: The sum of interior angles of a twelve-sided polygon (dodecagon) is = (12 – 2) x 180° = 1800°.
- Exterior angles: The sum of the exterior angles of a twelve-sided polygon (dodecagon) is 360°.
- Diagonals in dodecagon: 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
- Triangles in dodecagon: 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. It is given by:
Area = ½ × perimeter × apothem
The formula for area of a regular twelve-sided regular polygon (dodecagon) of side length d is given by:
Area = 3(2+√3)d^{2} ≈ 11.19615242 d^{2} |
The area calculated in terms of circumradius R of the circumscribed circle is;
Area = 3R^{2} |
Perimeter of Dodecagon
The perimeter is the total length of the boundaries of a twelve-sided polygon. The perimeter formula of dodecagon in terms of circumradius R is given by;
Perimeter = 12R√(2-√3) ≈ 6.2116570 R |
Facts on Dodecagon
Number of sides | 12 |
Number vertices | 12 |
Number of angles | 12 |
Interior angle | 150° |
Exterior angle | 30° |
Area | ½ × perimeter × apothem or 3(2+√3)d^{2} |
Perimeter | 12 × side |
Sum of interior angles | 1800° |
Related Articles
- Polygon
- Types Of Polygon
- Area Of Polygon
- Exterior Angles Of A Polygon
- Interior Angles of a Polygon
- Circumscribed
Solved Examples on Dodecagon
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)d^{2}
= 3(2+√3) x 10^{2}
= 11.19615242 x 100
Area ≈ 1119.615242 cm^{2}
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
Q.3: What is the perimeter of a regular dodecagon, that has a side length equal to 3.5 cm?
Solution: Given, the side-length = 3.5 cm
Perimeter = 12 x side = 12 x 3.5 = 42 cm
Practice Questions on Dodecagon
- Find the area of a dodecagon, whose each of its sides is equal to 5 cm.
- Find the perimeter of a dodecagon, whose each of its side is equal to 7cm.