In the world of geometry, shapes are everywhere, but have you ever stopped to wonder, “What exactly is a polygon?” Simply put, a polygon is a two-dimensional, flat shape that is closed and formed by straight line segments. Think of it as a figure drawn on a piece of paper where all sides are straight and they all connect to form a complete loop. These straight sides are also known as edges, and the points where these edges meet are called vertices or corners.
To get a clearer picture, let’s look at some examples of polygons:
- Triangles
- Squares
- Rectangles
- Pentagons
- Hexagons
These are all polygons because they are closed shapes made entirely of straight lines.
Now, to understand the definition even better, let’s see what is not a polygon. Shapes with curves or openings are not polygons.
Notice how the shapes above either have curved sides or aren’t fully closed. These characteristics disqualify them from being polygons.
Exploring the Polygon Family: Names and Classifications
Polygons are a diverse family of shapes, and we often categorize them based on their number of sides. The general way to name a polygon is by using the term “n-gon,” where “n” represents the number of sides. For instance, a polygon with five sides is a 5-gon, and one with ten sides is a 10-gon.
However, some polygons have special, more common names, especially those with fewer sides. The minimum number of sides a polygon can have is three. Why three? Because you need at least three straight lines to enclose an area and create a closed shape. Two lines can only form an angle or run parallel, but they can’t create a closed figure.
Here’s a handy chart to learn the names of polygons based on their sides:
While polygons with more than 10 sides also have specific names, they become quite complex and less frequently used. Therefore, using the n-gon naming convention is perfectly acceptable and often clearer for polygons with a higher number of sides.
Diving Deeper: Types of Polygons
Beyond the number of sides, polygons can be further classified based on their sides and angles. Let’s explore these different types:
1. Regular vs. Irregular Polygons: Side and Angle Equality
This classification hinges on whether the sides and angles within a polygon are equal or not.
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Regular Polygons: Imagine perfection in shape form! Regular polygons are those where all sides are of equal length, and all interior angles are of equal measure. They are symmetrical and balanced in appearance.
Examples of regular polygons include:
- Equilateral Triangle: A 3-sided regular polygon.
- Square: A 4-sided regular polygon.
- Regular Hexagon: A 6-sided regular polygon.
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Irregular Polygons: In contrast, irregular polygons are more diverse. These polygons have sides of different lengths and interior angles of different measures. They lack the perfect symmetry of regular polygons.
Examples of irregular polygons are shown below:
2. Convex vs. Concave Polygons: Angle Direction
This classification focuses on the direction of the interior angles and the diagonals of the polygon.
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Convex Polygons: Think of these as “bulging outwards.” A convex polygon is defined as a polygon where all interior angles are less than 180 degrees. Another way to visualize it is that if you take any two points inside a convex polygon, the line segment connecting them will always lie entirely within the polygon. Furthermore, all diagonals of a convex polygon reside within its boundaries.
Examples of convex polygons:
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Concave Polygons: These are polygons that “cave inwards” at least at one vertex. A concave polygon has at least one interior angle that is greater than 180 degrees (a reflex angle). If you draw a line segment between two points inside a concave polygon, parts of the segment might fall outside the polygon. Similarly, not all diagonals of a concave polygon are contained within it.
Examples of concave polygons:
Understanding the Difference:
3. Simple vs. Complex Polygons: Boundary Intersections
This classification is about how the sides of the polygon interact with each other.
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Simple Polygons: Simple polygons are, well, simpler in their structure! They have only one boundary, and their sides do not intersect each other. They form a single, non-self-intersecting loop. All the polygons discussed so far (regular, irregular, convex, concave) can be simple polygons.
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Complex Polygons (Self-Intersecting Polygons): Complex polygons are more intricate. They are polygons where sides cross over each other one or more times. This creates a shape that looks like it’s been folded or twisted.
Example of a complex polygon:
Angles Inside and Out: Sum of Angles in Polygons
Polygons are not just about sides; their angles are equally important! There are fascinating rules governing the sum of angles within polygons.
1. Sum of Interior Angles
The sum of all the interior angles inside a polygon depends on the number of sides it has. The formula to calculate this sum is:
(n – 2) × 180°
Where ‘n’ is the number of sides of the polygon.
For example, let’s take a hexagon (6 sides):
Sum of interior angles = (6 – 2) × 180° = 4 × 180° = 720°
So, if you add up all the interior angles of any hexagon, it will always total 720°.
2. Sum of Exterior Angles
Exterior angles are formed by extending one side of a polygon and measuring the angle between this extension and the adjacent side. Interestingly, the sum of the exterior angles of any polygon, regardless of the number of sides, is always 360°.
3. Angles in Regular Polygons: Equal Measures
In regular polygons, things become even simpler because both sides and angles are equal.
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Interior Angle of a Regular Polygon: To find the measure of each interior angle in a regular polygon, you first calculate the sum of interior angles using the formula (n-2) × 180°, and then divide this sum by the number of sides (n).
Each interior angle = ((n – 2) × 180°) / n
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Exterior Angle of a Regular Polygon: Similarly, for regular polygons, each exterior angle is also equal. Since the sum of exterior angles is always 360°, you just divide 360° by the number of sides (n) to find each exterior angle.
Each exterior angle = 360° / n
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Interior and Exterior Angle Relationship: At each vertex of any polygon (regular or irregular), the sum of the interior angle and the exterior angle is always 180°. They form a linear pair.
Solved Examples: Putting Polygon Knowledge to Practice
Let’s work through some examples to solidify your understanding of polygons.
Example 1: Fill in the blanks.
- The name of the three-sided regular polygon is an ________________.
- 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:
- equilateral triangle
- sides
- 360°
- line segments
Example 2: Match the polygon name with the number of sides.
- Nonagon a. 3
- Triangle b. 10
- Pentagon c. 9
- Decagon d. 5
Solution:
- c (9)
- a (3)
- d (5)
- b (10)
Example 3: Calculate the measure of each exterior angle of a regular polygon with 20 sides.
Solution:
For a regular polygon with n = 20 sides:
Each exterior angle = 360° / n = 360° / 20 = 18°
Example 4: The sum of the interior angles of a polygon is 1620°. How many sides does it have?
Solution:
Using the formula for the sum of interior angles: (n – 2) × 180° = 1620°
Divide both sides by 180°: n – 2 = 1620° / 180° = 9
Add 2 to both sides: n = 9 + 2 = 11
The polygon has 11 sides (an hendecagon or 11-gon).
Frequently Asked Questions About Polygons
What is a diagonal of a polygon?
A diagonal is a line segment that connects two non-adjacent vertices (corners) of a polygon. It’s essentially a line inside the polygon that is not a side.
What are the key properties of regular polygons?
Regular polygons are special because they have two main properties: all their sides are of equal length, and all their interior angles are of equal measure. This makes them highly symmetrical.
Is a circle a polygon?
No, a circle is not a polygon. Polygons are defined by having straight sides. A circle is a closed curve, but it has no straight sides. Therefore, it doesn’t fit the definition of a polygon.
What is the smallest number of sides a polygon can have?
The minimum number of sides for a polygon is three. A polygon with three sides is a triangle. You cannot create a closed shape with fewer than three straight line segments.
Can a polygon have a different number of angles than sides?
No, a polygon always has the same number of angles as it has sides. This is because each side connects to another side at a vertex, forming an angle. In a closed figure of straight lines, the number of vertices (angles) will always equal the number of sides.
