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.
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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:
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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.
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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:
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Irregular Polygons
Irregular polygons, on the other hand, have sides and angles of varying lengths. Here are some examples of irregular polygons:
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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:
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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:
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To understand the difference between convex and concave polygons, have a look at the image below:
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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:
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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:
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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:
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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:
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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.
