Regular Polygon – Definition With Examples
Updated on January 14, 2024
Welcome to Brighterly, the home of engaging and interactive learning! Today, let’s delve into the captivating world of geometry, and specifically, let’s explore the concept of a Regular Polygon. A regular polygon is not just any shape; it’s a perfect blend of equality and symmetry, a testament to the harmony that mathematics can bring into our world.
A regular polygon is a geometric marvel, a flat, closed figure created with straight lines. What sets it apart is its special characteristic: each of its sides and angles are equal. Imagine a square or an equilateral triangle. Every side of the square is identical in length, and every angle inside it mirrors the others in magnitude. The same applies to the equilateral triangle, each of its sides is identical, and all its internal angles match.
These shapes, along with many others, belong to the family of regular polygons. They are found everywhere in our surroundings, from the patterned tiles on the floor to the intricate designs of snowflakes. By understanding regular polygons, we open doors to understanding the mathematical harmony that underlies our world.
What is a Regular Polygon?
A regular polygon is a geometric shape that is flat, straightsided, and closed. It is a special type of polygon where all its sides and angles are equal. Consider a square or an equilateral triangle, they are perfect examples of a regular polygon. Every side of a square is the same length, and every angle inside it is exactly the same as the others. Similarly, for an equilateral triangle, each of its sides is of the same length and all its internal angles are equal. Some examples of regular polygons include squares, equilateral triangles, regular pentagons, and regular hexagons. Each of these shapes is unique, yet they all share common properties that define them as regular polygons.
What Is a Polygon?
Before we delve deeper into regular polygons, let’s take a step back and understand what a polygon is. A polygon is a 2dimensional shape made up of straight lines. It is a closed figure, meaning its lines connect at every end. It can have any number of sides, as long as it has at least three. Triangles, squares, and rectangles are all polygons. However, not all polygons are regular.
Parts of a Polygon
A polygon consists of several parts including the sides, vertices, and angles. The sides are the straight lines that form the boundary of the polygon. The points where the sides meet are called vertices. The angles are the spaces between the sides, found inside the polygon at the vertices.
Properties of Regular Polygons
Regular polygons have some unique properties. Firstly, all their sides and angles are equal. Secondly, they are symmetrical around their center, meaning if you were to draw lines from the center to each vertex, the shape would look the same from any direction. Lastly, they have a constant radius, which is the distance from the center to any vertex.
Perimeter of a Regular Polygon
The perimeter of a regular polygon can be calculated by multiplying the length of one side by the number of sides. For instance, if a regular hexagon (a sixsided regular polygon) has each side measuring 5 units, its perimeter would be 6 * 5 = 30 units.
Sum of Interior Angles of a Regular Polygon
The sum of the interior angles of a regular polygon can be found using the formula (n2) x 180°
, where n
is the number of sides. For instance, a square (4 sides) has an interior angle sum of (42) x 180°
= 360°.
Measure of Each Interior Angle of a Regular Polygon
The measure of each interior angle of a regular polygon is given by the formula [(n2) x 180°]/n
, where n
is the number of sides. For example, each interior angle of a square (4 sides) is [(42) x 180°]/4
= 90°.
Measure of Each Exterior Angle of a Regular Polygon
The measure of each exterior angle of a regular polygon is calculated by the formula 360°/n
, where n
is the number of sides. For a square (4 sides), each exterior angle measures 360°/4
= 90°.
Number of Diagonals of a Regular Polygon
The number of diagonals of a regular polygon can be calculated using the formula n(n3)/2
, where n
is the number of sides. For instance, a pentagon (5 sides) has 5(53)/2
= 5 diagonals.
Number of Triangles of a Regular Polygon
We can divide any regular polygon into triangles by drawing lines from one vertex to all other vertices. The number of such triangles is n2
, where n
is the number of sides. For a hexagon (6 sides), we can form 62
= 4 triangles.
Lines of Symmetry of a Regular Polygon
The number of lines of symmetry in a regular polygon is equal to the number of its sides. A square (4 sides) has 4 lines of symmetry, while an equilateral triangle (3 sides) has 3 lines of symmetry.
Order of Symmetry of a Regular Polygon
The order of symmetry of a regular polygon is the number of times the shape maps onto itself as it rotates from 0° to 360°. This is also equal to the number of sides. A square (4 sides), for instance, has an order of symmetry of 4.
Different Regular Polygons
There are numerous types of regular polygons, each with its unique properties. Some common ones include triangles, squares, pentagons, hexagons, heptagons, octagons, nonagons, and decagons. The number of sides can technically be infinite, leading to a shape called a circle!
Solved Examples on Regular Polygon
Let’s consider a few solved examples to understand the concept of regular polygons better:

Example 1: Find the perimeter of a regular pentagon with each side measuring 7 cm. Solution: Perimeter = number of sides x length of one side = 5 x 7 = 35 cm.

Example 2: Calculate the sum of the interior angles of a regular octagon. Solution: Sum of interior angles = (n2) x 180° = (82) x 180° = 1080°.
Practice Problems on Regular Polygon
Now, it’s your turn to apply your newfound knowledge. Here are a few practice problems for you:
 What is the measure of each interior angle of a regular hexagon?
 How many diagonals does a regular decagon have?
 How many lines of symmetry does a regular pentagon have?
Conclusion
As we reach the end of our engaging journey through the realm of regular polygons, it’s evident that these shapes are much more than just figures with equal sides and angles. They embody beauty, symmetry, and balance. At Brighterly, we hope that this deep dive into regular polygons has sparked a fascination for the intricate beauty that mathematics holds.
Regular polygons form the foundation of more complex geometric studies, allowing us to comprehend the world in a more structured and symmetrical way. Architects use the properties of regular polygons to design sturdy and aesthetically pleasing structures. Designers use them to create patterns that are pleasing to the eye. Engineers use them to simplify complex problems and find efficient solutions.
But it’s not just in these professional fields that regular polygons make their mark. They are all around us – in the design of a soccer ball, the structure of a beehive, the symmetry of a snowflake, and many more. With each side and angle echoing the other, regular polygons demonstrate the harmony that can exist in complexity.
So the next time you come across a shape with equal sides and angles, remember it’s not just a shape, it’s a regular polygon, a testament to the harmony and balance in our universe. And as you continue to explore the fascinating world of geometry, remember that Brighterly is here to light up your path, making learning brighter and more enjoyable for you!
Frequently Asked Questions on Regular Polygon
What is the difference between a regular and an irregular polygon?
A regular polygon is a polygon with all sides and angles equal. In contrast, an irregular polygon has sides and angles that are not equal. For example, a square is a regular polygon because all its sides and angles are equal, while a trapezoid is an irregular polygon because its sides and angles are not equal.
How can I determine if a polygon is regular or irregular?
To determine if a polygon is regular, check if all its sides have equal length and all its interior angles are equal. If the sides and angles are not equal, the polygon is irregular. You can use a ruler to measure the sides and a protractor to measure the angles to verify if they are equal.
Can a polygon have curved sides?
No, a polygon cannot have curved sides. By definition, a polygon is a 2dimensional shape made up of straight lines. If a shape has curved sides, it is not considered a polygon.
What is the smallest possible number of sides a regular polygon can have?
The smallest possible number of sides a regular polygon can have is three. A polygon with three sides is called a triangle. An equilateral triangle, with all sides and angles equal, is an example of a regular polygon.
Can a regular polygon have an odd number of sides?
Yes, a regular polygon can have an odd number of sides. For example, a regular pentagon has five sides, and a regular heptagon has seven sides. As long as all the sides and angles are equal, the polygon is considered regular, regardless of whether the number of sides is odd or even.
How are the diagonals of a regular polygon related to its symmetry?
The diagonals of a regular polygon can be drawn from one vertex to all other nonadjacent vertices. In the case of a regular polygon, the diagonals also act as lines of symmetry, dividing the polygon into congruent sections.
What happens to the shape of a regular polygon as the number of sides increases?
As the number of sides in a regular polygon increases, its shape becomes more circular. With an infinite number of sides, a regular polygon would essentially become a circle. The circle can be thought of as the limiting case of a regular polygon as the number of sides approaches infinity.