Are you curious about the fascinating world of polygons? In geometry, a polygon is a two-dimensional shape with straight sides that form a closed figure. Unlike curved shapes, polygons have straight sides and defined corners called vertices. Let's explore the different types and properties of polygons to deepen our understanding.

## What Defines a Polygon?

A polygon is a flat, closed shape with straight sides. It is important to note that curved sides are not considered as part of a polygon. Take a look at the image below to visualize the sides/edges of a polygon.

*Image source: sanaulac.vn*

While polygons come in various shapes and sizes, they all share this common characteristic of having straight sides. Now, let's explore some examples of polygons.

## Examples of Polygons

Here are a few examples of polygons:

*Image source: sanaulac.vn*

As you can see, these shapes have closed boundaries and straight sides, making them polygons. However, it's essential to understand the distinction between polygons and non-polygons.

## Non-Examples of Polygons

Not all closed shapes with straight sides are considered polygons. Let's take a look at some non-examples of polygons.

*Image source: sanaulac.vn*

Although these shapes may have straight sides, they either have open boundaries or contain curved sections. Remember, polygons must have closed boundaries and only consist of straight sides.

## Understanding Polygon Classification

Polygons can be classified based on the number of sides and angles they possess. Let's explore the two primary classifications:

## Classification Based on Sides: Regular and Irregular Polygons

### Regular Polygons

Regular polygons are polygons with equal sides and angles. Each side of a regular polygon has the same length, and every angle within the shape is identical. Here are a few examples of regular polygons:

*Image source: sanaulac.vn*

### Irregular Polygons

Irregular polygons, on the other hand, have sides and angles of varying lengths. Here are some examples of irregular polygons:

*Image source: sanaulac.vn*

## Classification Based on Angles: Convex and Concave Polygons

### Convex Polygons

A convex polygon is a polygon in which all interior angles are less than 180°. In a convex polygon, all diagonals (line segments connecting non-consecutive vertices) lie within the interior of the shape. Take a look at some examples of convex polygons:

*Image source: sanaulac.vn*

### Concave Polygons

Concave polygons, on the other hand, have at least one interior angle greater than 180°. In a concave polygon, not all diagonals lie within the interior of the shape. Here are some examples of concave polygons:

*Image source: sanaulac.vn*

To understand the difference between convex and concave polygons, have a look at the image below:

*Image source: sanaulac.vn*

## Simple and Complex Polygons

### Simple Polygons

A simple polygon has only one boundary, and its sides do not intersect. Here's an example of a simple polygon:

*Image source: sanaulac.vn*

### Complex Polygons

A complex polygon, on the other hand, has sides that intersect one another one or more times. This intersection creates a more intricate shape. Here's an example of a complex polygon:

*Image source: sanaulac.vn*

## Understanding the Angles of a Polygon

The angles of a polygon play a crucial role in determining its properties. Let's explore the sum of the interior and exterior angles of a polygon.

### Sum of Interior Angles of a Polygon

The sum of the interior angles of a polygon with n sides can be calculated using the formula: (n - 2) × 180°. Here's an example:

*Image source: sanaulac.vn*

In this example, the sum of the interior angles of a polygon with 6 sides is (6 - 2) × 180° = 720°. Each interior angle can be calculated by dividing the sum by the number of sides.

### Sum of Exterior Angles of a Polygon

The sum of the exterior angles of any polygon is always 360°, regardless of the number of sides it has. Each exterior angle can be calculated by dividing 360° by the number of sides. Take a look at the example below:

*Image source: sanaulac.vn*

In this example, the sum of the exterior angles of a polygon with 5 sides is 360°. Each exterior angle is equal to 360° divided by the number of sides.

## Angles in a Regular Polygon

In a regular polygon, all sides, interior angles, and exterior angles are equal. Let's take a closer look at the properties of interior and exterior angles in regular polygons.

### Interior Angle

The sum of the interior angles of a regular polygon with n sides can be calculated using the formula: (n - 2) × 180°. Each interior angle can be calculated by dividing the sum by the number of sides.

### Exterior Angle

The sum of the exterior angles of any polygon is always 360°, regardless of the number of sides. Each exterior angle is equal to 360° divided by the number of sides.

### Sum of Interior and Exterior Angles

In both regular and irregular polygons, the sum of an interior angle and an exterior angle at each vertex is always 180°.

## Solving Examples on Polygons

Let's solve a few examples to solidify our understanding of polygons.

### Example 1: Fill in the blanks.

- The name of the three-sided regular polygon is
.**____** - A regular polygon is a polygon whose all _
are equal, and all angles are equal.**____** - The sum of the exterior angles of a polygon is
.**__** - A polygon is a simple closed figure formed by only _
.**__**

**Solution:**

- The three-sided regular polygon is called an
**equilateral triangle**. - A regular polygon is a polygon whose all
**sides**are equal, and all angles are equal. - The sum of the exterior angles of a polygon is
**360°**. - A polygon is a simple closed figure formed by only
**line segments**.

### Example 2: Write the number of sides for a given polygon.

**Nonagon****Triangle****Pentagon****Decagon**

**Solution:**

- A nonagon has
**9**sides. - A triangle has
**3**sides. - A pentagon has
**5**sides. - A decagon has
**10**sides.

### Example 3: Find the measure of each exterior angle of a regular polygon with 20 sides.

**Solution:**

The polygon has 20 sides, so n = 20.

The sum of the exterior angles of a polygon is 360°.

So, each exterior angle = 360°/n = 360°/20 = 18°.

Each exterior angle of the regular polygon with 20 sides measures **18°**.

### Example 4: The sum of the interior angles of a polygon is 1620°. How many sides does it have?

**Solution:**

The sum of the interior angles of a polygon with n sides can be calculated using the formula: (n - 2) × 180°.

Given that the sum of the interior angles is 1620°, we can set up the equation:

(n - 2) × 180° = 1620°

Simplifying the equation:

(n - 2) = 1620°/180°

(n - 2) = 9

n = 9 + 2

n = 11

So, the polygon has **11** sides.

## Practice Problems on Polygons

Now that you have a solid understanding of polygons, why not test your knowledge with practice problems? Take a look at the following exercises to further enhance your skills.

## Frequently Asked Questions about Polygons

- What defines a polygon?
- How are polygons classified?
- What are the properties of regular polygons?
- How are the interior and exterior angles of a polygon calculated?

## Conclusion

Polygons are fascinating shapes with unique properties. By understanding the types, angles, and classifications of polygons, you can deepen your geometric knowledge. Remember, whether you're exploring regular, irregular, convex, or concave polygons, each shape has its distinct characteristics. So, dive into the exciting world of polygons and unlock the secrets of their boundless geometrical possibilities.