Dodecagon

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.

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 12 sides of equal length and all angles of equal measure. 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 and angles of different measures. 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

In a concave dodecagon, at least one of its interior angles is greater than 180° and the vertices of a concave dodecagon can be inwards and outwards as seen in the figure given below.

Convex Dodecagon

A dodecagon in which none of its interior angles is greater than 180° and where no vertex points inwards is called a convex dodecagon as seen in the figure given below.

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 (180°). 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 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 that 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.

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.

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 Tips on Dodegaon

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 ) × s²
• The perimeter of a dodecagon is calculated with the formula: s × 12.

☛ Related Articles

• Types of Polygon
• Pentagon
• Hexagon
• Heptagon
• Octagon
• Decagon
• Dodecahedron

What is a Dodecagon?

A dodecagon is a polygon with 12 sides, 12 angles, and 12 vertices. Dodecagons can be of different types depending upon the measure of their sides, angles, and many such properties. A regular dodecagon has all 12 sides of equal length and all angles of equal measure. An irregular dodecagon has sides and angles of different measures.

What is the Sum of Exterior Angles of a Dodecagon?

The sum of the exterior angles of any polygon is always equal to 360° irrespective of the number of sides. Therefore, even in a dodecagon, the sum of the exterior angles is 360°.

What is the Sum of Interior Angles of a Dodecagon?

The sum of the interior angles of a dodecagon can be calculated with the help of the formula: (n - 2 ) × 180°. Here, 'n' represents the number of sides of any polygon. In this case, n = 12. So, after substituting the value of 'n' in the formula: (12 - 2) × 180° = 1800°. Therefore, the sum of interior angles of a dodecagon is equal to 1800°.

What are the Properties of Regular Dodecagons?

In a regular decagon, each interior angle is equal to 150° and each exterior angle is equal to 30°. The sum of its interior angles is 1800°, while the sum of its exterior angles is 360°.

What is the Area of a Dodecagon?

The area of a dodecagon is calculated with the help of the formula: A = 3 × ( 2 + √3 ) × s², where s = the length of the side of the dodecagon.

What is One Exterior Angle of a Dodecagon?

Each exterior angle of a dodecagon is equal to 30°. If we observe a dodecagon, we can see that the exterior angle and interior angle form a straight line. We know that each interior angle of a dodecagon is 150°. Since they together form a linear pair of angles, each exterior angle is 180° - 150° = 30°.

How Many Sides are there in a Dodecagon?

There are 12 sides in a dodecagon. The word dodecagon comes from the Greek word "dōdeka" which means 12 and "gōnon" which means angle.

How Many Triangles are there in a Dodecagon?

A dodecagon can be split into a series of triangles by the diagonals that are drawn from its vertices. The number of triangles that are formed 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.

How to Find the Perimeter of a Dodecagon?

The perimeter of a dodecagon is calculated with the formula: P = s × 12, where s = length of one side of the dodecagon. For example, the length of one side of a dodecagon is 20 units. So, its perimeter will be: 20 × 12 = 240 units.

How Many Diagonals are there in a Dodecagon?

There are 54 diagonals in a dodecagon. These diagonals can be calculated with the help of 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.

Is a Dodecagon a Regular Polygon?

A dodecagon can be a regular polygon if all its 12 sides are of equal length and the interior angles are of equal measure.