A dodecagon is a polygon with 12 sides, 12 angles, and 12 vertices. The word dodecagon comes from the Greek word "dōdeka" which means 12 and "gōnon" which means angle. This polygon can be regular, irregular, concave, or convex, depending on its properties.

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 1 What is a Dodecagon? 2 Types of Dodecagons 3 Properties of a Dodecagon 4 Perimeter of a Dodecagon 5 Area of a Dodecagon 6 FAQs on Dodecagon

A dodecagon is a 12-sided polygon that encloses space. Dodecagons can be regular in which all interior angles and sides are equal in measure. They can also be irregular, with different angles and sides of different measurements. The following figure shows a regular and an irregular dodecagon. Dodecagons can be of different types depending upon the measure of their sides, angles, and many such properties. Let us go through the various types of dodecagons.

Regular Dodecagon

A regular dodecagon has all the 12 sides of equal length, all angles of equal measure, and the vertices are equidistant from the center. It is a 12-sided polygon that is symmetrical. Observe the first dodecagon shown in the figure given above which shows a regular dodecagon.

Irregular Dodecagon

Irregular dodecagons have sides of different shapes and angles.There can be an infinite amount of variations. Hence, they all look quite different from each other, but they all have 12 sides. Observe the second dodecagon shown in the figure given above which shows an irregular dodecagon.

Concave Dodecagon

A concave dodecagon has at least one line segment that can be drawn between the points on its boundary but lies outside of it. It has at least one of its interior angles greater than 180°. Convex Dodecagon

A dodecagon where no line segment between any two points on its boundary lies outside of it is called a convex dodecagon. None of its interior angles is greater than 180°. ## Properties of a Dodecagon

The properties of a dodecagon are listed below which explain about its angles, triangles and its diagonals.

Interior Angles of a Dodecagon

Each interior angle of a regular dodecagon is equal to 150°. This can be calculated by using the formula:

$$\frac{180n–360} {n}$$, where n = the number of sides of the polygon. In a dodecagon, n = 12. Now substituting this value in the formula.

\begin{align} \frac{180(12)–360} {12} = 150^\circ \end{align}

The sum of the interior angles of a dodecagon can be calculated with the help of the formula: (n - 2 ) × 180° = (12 – 2) × 180° = 1800°. Exterior Angles of a Dodecagon

Each exterior angle of a regular dodecagon is equal to 30°. If we observe the figure given above, we can see that the exterior angle and interior angle form a straight angle. Therefore, 180° - 150° = 30°. Thus, each exterior angle has a measure of 30°. The sum of the exterior angles of a regular dodecagon is 360°.

Diagonals of a Dodecagon

The number of distinct diagonals that can be drawn in a dodecagon from all its vertices can be calculated by using the formula: 1/2 × n × (n-3), where n = number of sides. In this case, n = 12. Substituting the values in the formula: 1/2 × n × (n-3) = 1/2 × 12 × (12-3) = 54

Therefore, there are 54 diagonals in a dodecagon.

Triangles in a Dodecagon

A dodecagon can be broken into a series of triangles by the diagonals which are drawn from its vertices. The number of triangles which are created by these diagonals, can be calculated with the formula: (n - 2), where n = the number of sides. In this case, n = 12. So, 12 - 2 = 10. Therefore, 10 triangles can be formed in a dodecagon.

The following table recollects and lists all the important properties of a dodecagon discussed above.

 Properties Values Interior angle 150° Exterior angle 30° Number of diagonals 54 Number of triangles 10 Sum of the interior angles 1800°

## Perimeter of a Dodecagon

The perimeter of a regular dodecagon can be found by finding the sum of all its sides, or, by multiplying the length of one side of the dodecagon with the total number of sides. This can be represented by the formula: P = s × 12; where s = length of the side. Let us assume that the side of a regular dodecagon measures 10 units. Thus, the perimeter will be: 10 × 12 = 120 units.

## Area of a Dodecagon

The formula for finding the area of a regular dodecagon is: A = 3 × ( 2 + √3 ) × s2 , where A = the area of the dodecagon, s = the length of its side. For example, if the side of a regular dodecagon measures 8 units, the area of this dodecagon will be: A = 3 × ( 2 + √3 ) × s2 . Substituting the value of its side, A = 3 × ( 2 + √3 ) × 82 . Therefore, the area = 716.554 square units.

Important Notes

The following points should be kept in mind while solving problems related to a dodecagon.

Dodecagon is a 12-sided polygon with 12 angles and 12 vertices.The sum of the interior angles of a dodecagon is 1800°.The area of a dodecagon is calculated with the formula: A = 3 × ( 2 + √3 ) × s2The perimeter of a dodecagon is calculated with the formula: s × 12.

## Related Articles on Dodecagon

Check out the following pages related to a dodecagon.

Example 1: Identify the dodecagon from the following polygons. Solution:

A polygon with 12 sides is known as a dodecagon. Therefore, figure (a) is a dodecagon.

Example 2: There is an open park in the shape of a regular dodecagon. The community wants to buy a fencing wire to place it around the boundary of the park. If the length of one side of the park is 100 meters, calculate the length of the fencing wire required to place all along the park's borders.

Solution:

Given, the length of one side of the park = 100 meters. The perimeter of the park can be calculated using the formula: Perimeter of a dodecagon = s × 12, where s = the length of the side. Substituting the value in the formula: 100 × 12 = 1200 meters.

Therefore, the length of the required wire is 1200 meters.

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Example 3: If each side of a dodecagon is 5 units, find the area of the dodecagon.