# Dodecagon

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 1 Introduction to Dodecagon 2 Thinking Out of the Box! 3 Properties of Dodecagon 4 Perimeter of Dodecagon 5 Area of Dodecagon 6 Important notes on Dodecagon 7 Solved Examples on Dodecagon 8 Practice Questions on Dodecagon 9 Maths Olympiad Sample Papers 10 Frequently Asked Questions (FAQs)

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## Introduction to Dodecagon

### Dodecagon definition:

A dodecagon is a polygon with 12 sides.

A dodecagon has 12 angles and 12 vertices.

The word dodecagon comes from the Greek word "dōdeka" which means 12 and "gōnon" which means angle.

A dodecagon in Geometry is a type of polygon with the following properties.

1. It is a two-dimensional figure.
2. It consists of 12 straight sides enclosing a space.
3. It has 12 interior angles.

### Regular Dodecagon

All the 12 sides of a regular dodecagon are of equal length.

The vertices are equidistant from the center.

A regular dodecagon is a 12-sided polygon that is symmetrical.

### Irregular Dodecagon

They are dodecagons with sides of different shapes and angles.

There can be an infinite amount of variations.

Hence, they all look very different from each other, but they all have 12 sides.

Now, let us explore the difference between a regular dodecagon and an irregular dodecagon using a simulation.

Alter the dodecagon sides in the simulation.

You can view how an irregular dodecagon is formed from a regular dodecagon.

### Concave Dodecagon

A dodecagon that has at least one line segment that can be drawn between points on its boundary but lies outside of it is called a concave dodecagon.

It has at least one of its interior angles greater than $$180\ ^\circ$$.

Concave Dodecagon

### 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 are greater than $$180\ ^\circ$$.

Convex Dodecagon

Think Tank
1. Imagine you are given a project to arrange all the zodiac signs and their information in one chart. How would you arrange it creatively?

(HINT: Regular Dodecagon)

## Properties of Dodecagon

### Angles of Dodecagon:

A dodecagon can be broken into a series of triangles by diagonals which are drawn from its vertices.

From these triangles, we can find the sum of the interior angles.

### Interior angle:

Since the sum of the degrees in a triangle is $$180\ ^\circ$$, the sum of the interior angles of a dodecagon is $$10 \times 180\ ^\circ = 1800\ ^\circ$$

Since $$1880\ ^\circ \div 12 = 150\ ^\circ$$, each interior angle in a regular dodecagon has a measure of $$150\ ^\circ$$

We can also find the measure of each interior angle in a dodecagon using the following formula:

For a dodecagon, n=12

$\frac{180n–360} {n}$

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

Since the exterior angle and interior angle form a straight angle, we have

\begin{align} 180^\circ – 150^\circ = 30^\circ \end{align}

Thus, each exterior angle has a measure of $$30^\circ$$.

### Dodecagon Triangles

The number of triangles in a dodecagon created by drawing diagonals from a given vertex is:

 $$n - 2 = (12-2)= 10$$

### Dodecagon Diagonals

The number of distinct diagonals that can be drawn from all vertices can be calculated using the formula:

 $$\frac{1}{2}n(n-3)$$

\begin{align}\frac{1}{2}n(n-3) = \frac{1}{2}12(12-3)= 54 \end{align}

According to dodecagon geometry, there are 54 diagonals in a dodecagon.

The following table lists all the important properties of a dodecagon.

Properties Values

Interior angle

$$150^\circ$$

Exterior angle

$$30^\circ$$

Number of diagonals

54

Number of triangles

10

Sum of the interior angles

$$1880^\circ$$

## Perimeter of Dodecagon

The perimeter of a regular dodecagon can be found by multiplying the length of the dodecagon side with the number of sides.

 $$X \times n$$

Where,

• X = Length of the side
• n = Number of sides

Let us now apply this formula in an example.

Let's assume that the side of a regular dodecagon measures 10 cm.

Then the perimeter will be:

\begin{align} 10 \times 12 = 120\:cm \end{align}

## Area of Dodecagon

The formula for finding the area of a regular dodecagon is:

### Area of Dodecagon (A):

 $$A = 3\times(2+√3)\times s²$$

where, $$A$$ is the area of a dodecagon and $$s$$ is the length of a side.

You can calculate the area of a regular dodecagon using the following simulation.

Vary the length of the sides to observe the difference in areas.

### Finding the area of an irregular dodecagon:

There is no fixed formula for finding the area of an irregular dodecagon as the sides and angles are not equal.

We can calculate the area by dividing the figure into regular shapes, finding those areas separately and adding them all up.

Important Notes
1. Dodecagon is a 12-sided polygon with 12 angles and 12 vertices.
2. Dodecagon is classified into regular, irregular, concave and convex.
3. The sum of the interior angles of a dodecagon is $$1800^\circ$$
4. Perimeter of a dodecagon is $$s\times 12$$
5. Area of a dodecagon is $$A = 3\times(2+√3)\times s²$$
6. In English alphabets, capitals of the letters H, X, and E have dodecagonal outlines.
7. The regular dodecagon features prominently in many buildings. The Torre del Oro is a dodecagonal military watchtower in Seville, southern Spain, built by the Almohad dynasty.

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## Solved Examples on Dodecagon

 Example 1

Identify the dodecagon from the following polygons.

Solution:

A polygon with 12 sides is known as a dodecagon.

 $$\therefore$$ (a) is a dodecagon.
 Example 2

Which category of dodecagon does this architecture belong to?

(a) Irregular and concave dodecagon

(b) Regular and convex dodecagon

Solution:

The architecture in the image has a dodecagon shape.

The tower can be categorized as a regular and convex dodecagon.

 $$\therefore$$ (b) Regular and convex dodecagon.
 Example 3

If each side of a dodecagon is 5 cm, then find the area of the dodecagon.

Solution:

Given that, a = 5 cm

 $$A = 3\times(2+√3)\times s²$$

Area of dodecagon = $$3 (2 + √3) a^2$$

= $$3 (2 + √3) 5^2$$

= $$75 (2 + √3)\:cm^2$$

(By substituting the value of √3 = 1.732)
= $$279.75\:cm^2$$

 $$\therefore$$ Area of Dodecagon = $$279.75\:cm^2$$
 Example 4

There is an open park which has a regular dodecagon shape.

The community wants to put up a fence along its borders to keep the children safe at the park.

They need to buy fencing wire to place along the perimeter of the park.

If the length of one side of the park is 100 meters, calculate the length of 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 find using the formula:

 $$X \times n$$

Perimeter of dodecagon = $$X \times n$$

We know that n = 12 and X = 100 meters

Thus,

\begin{align} 100 \times 12 = 1200\: meters \end{align}

Therefore, the total fencing wire required is 1200 meters.

 $$\therefore$$ Perimeter = 1200 meters
 Example 5

A couple of years ago, Hong Kong minted two-dollar coins in the shape of dodecagonal scallops.

Assuming one side of this beautiful coin is 6.278 mm in length, what would roughly be the area of this coin?

Solution:

The area of the coin can be calculated using the formula for the area of a dodecagon.

 $$A = 3\times(2+√3)\times s²$$

\begin{align}A = 3 × (s)^2 × (2 + √3)\end{align}
\begin{align}A = 3 × (6.278\: mm)^2 \times (2 + √3)\end{align}
\begin{align}A = 118.239852\: mm^2 \times 3.73205080757\end{align}
\begin{align}A = 441.277135143\: mm^2\end{align}
\begin{align}A = 4.41277 \:cm^2\end{align}

 $$\therefore$$ Area of the coin = $$4.41277\:cm^2$$

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## Practice Questions on Dodecagons

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

## 1. Is a dodecagon a regular polygon?

A dodecagon is a 12 sided polygon with regular and irregular geometry.

## 2. What are the properties of regular dodecagons?

• All 12 sides are of equal length.
• The vertices are equidistant from the center.
• The sides and angles are symmetrical.

## 3. What are some of the important properties of dodecagons?

• Interior angle -  $$150^\circ$$
• Exterior angle - $$30^\circ$$
• Number of dodecagon diagonals - 54
• Number of triangles - 10
• Sum of interior angles - $$1880^\circ$$

## 4.How do you find the area of a dodecagon?

The area of a dodecagon is calculated using the formula: $$A = 3\times(2+√3)\times s²$$, where $$s$$ is the length of the side of the dodecagon.

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