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).
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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.
Types of Dodecagon
There are basically four types of a dodecagon, they are:
- Regular dodecagon
- Irregular dodecagon
- Convex dodecagon
- Concave 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 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.
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)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. 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 of angles||12|
|Area||½ × perimeter × apothem or 3(2+√3)d2|
|Perimeter||12 × side|
|Sum of interior angles||1800°|
- Types Of Polygon
- Area Of Polygon
- Exterior Angles Of A Polygon
- Interior Angles of a Polygon
Solved Examples on Dodecagon
Q. 1: Calculate the area of the dodecagon with side length d = 10 cm.
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
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
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.