What is a Regular Polygon | Definitions, Properties and Examples

Regular Polygons are closed two-dimensional planar figures made of straight lines having equal sides and equal angles. These symmetrical shapes, ranging from equilateral triangles to perfect decagons, They are important in various fields such as architecture, art, and design.

The sides or edges of a polygon are formed by linking end-to-end segments of a straight line to form a closed shape. The intersection of two line segments that result in an angle is referred to as a vertex or corner. A polygon is referred to be a regular polygon if all of its sides are congruent.

In this article, we will discuss Regular Polygon its types and properties along with some solved examples and questions on regular polygon.

Regular Polygon Definition

Regular polygons are closed symmetric figures made with straight lines with all sides and angles equal. This symmetry gives them a balanced and uniform appearance, making them an essential concept in geometry.

Squares, rhombuses, equilateral triangles, and other shapes serve as cases of regular polygons. They have both congruent angles and congruent sides. They are equiangular shapes.

Parts of Polygon

A polygon has 3 parts:

• Angles: Angles are the geometric figures formed by joining the sides of the polygon.
  • Interior Angle: Angles within the enclosed area of the polygon
  • Exterior Angle: Angles formed outside the polygon by extending one side.
• Sides: A line segment that joins two vertices is known as a side.
• Vertices: The point at which two sides meet is known as a vertex.

Regular Polygons Examples

There are many examples of regular polygons in real life around us. A common example is a stop sign, an octagon with eight equal sides. Road signs often employ triangles, squares, and pentagons too. Nature showcases these shapes in beehives (hexagons) and snowflakes (hexagrams). Regular polygons' uniformity and symmetry make them prevalent in architecture, such as the facades of buildings and decorative tiles.

The image added below shows regular and irregular polygons.

Fun Fact : The equal sides ensure that regular polygons can perfectly fit within circumscribed and inscribed circles, further demonstrating their geometric perfection.

Regular Polygon Shapes

There are various regular polygons, such as equilateral triangles, squares, regular pentagons, etc. There can be any number of regular polygons based on the number of sides they have. If it has three sides, it is an equilateral triangle. If it has four sides, it is a square. If it has five sides, it is a regular pentagon, and so on. Let's discuss these individual shapes:

Equilateral Triangle

Properties of Equilateral Triangle

• 3 Equal sides, 3 Vertices, 3 Angles.
• The sum of the interior angle of an equilateral triangle is 180°
• Each interior angle is 60° and each exterior angle is 120°.
• No of Diagonals : 0.
• No of triangle formed : 1
• Axis of symmetry : 3

Square

Properties of Square

• 4 Equal sides, 4 Vertices, 4 Angles.
• The sum of the interior angle of an equilateral triangle is 360°
• Each interior angle is 90° and each exterior angle is 90°.
• No of Diagonals : 2.
• No of triangle formed : 2
• Axis of symmetry : 4

Pentagon

Properties of Pentagon

• 5 Equal sides, 5 Vertices, 5 Angles.
• The sum of the interior angle of an equilateral triangle is 540°
• Each interior angle is 108° and each exterior angle is 72°.
• No of Diagonals : 5.
• No of triangle formed : 3
• Axis of symmetry : 5

Hexagon

Properties of Hexagon

• 6 Equal sides, 6 Vertices, 6 Angles.
• The sum of the interior angle of an equilateral triangle is 720°
• Each interior angle is 120° and each exterior angle is 60°.
• No of Diagonals : 9.
• No of triangle formed : 4
• Axis of symmetry : 6

Heptagon

Properties of Heptagon

• 7 Equal sides, 7 Vertices, 7 Angles.
• The sum of the interior angle of an equilateral triangle is 900°
• Each interior angle is 128.57° and each exterior angle is 51.43°.
• No of Diagonals : 14.
• No of triangle formed : 5
• Axis of symmetry : 7

Octagon

Properties of Octagon

• 8 Equal sides, 8 Vertices, 8 Angles.
• The sum of the interior angle of an equilateral triangle is 1080°
• Each interior angle is 135° and each exterior angle is 45°.
• No of Diagonals : 20.
• No of triangle formed : 6
• Axis of symmetry : 8

Properties of Regular Polygons

The general properties of the Regular Polygons are discussed below,

Sum of Interior Angles of a Regular Polygon

Sum of Interior Angles of a Regular Polygon is given using the formula,

Sum of Interior Angles = 180°(n - 2)

where "n" represents the number of sides of a regular polygon

Each Interior Angle of a Regular Polygon

Each Interior angle of an n-sided regular polygon is measured using the formula,

Each Interior Angle = [(n - 2) x 180°]/n

where "n" represents the number of sides of a regular polygon

Exterior Angle of a Regular Polygon

Each Exterior Angle of an n-sided regular polygon is measured using the formula,

Each Exterior Angle =360°/n

where "n" represents the number of sides of a regular polygon

Number of Diagonals in a Regular Polygon

Number of diagonals in an n-sides polygon is given using the formula,

Number of Diagonals = n(n - 3)/2

where "n" represents the number of sides of a regular polygon

Number of Triangles in a Regular Polygon

Number of Triangles that can be generated inside by connecting diagonals of a Regular n - sided Polygon is given using this formula

Number of triangles = (n - 2)

where "n" represents the number of sides of a regular polygon

Number of Axis of symmetry in a Regular Polygon

The number of Axis of Symmetry in an n-sides polygon (imaginary line dividing a shape into two equal halves)

number of Axis of Symmetry = n

where "n" represents the number of sides of a regular polygon

Regular Polygons Formulas

Regular polygons are two-dimensional closed figures with finite straight lines, as we have explained. It is made up of straight lines that join. The formulas used in a regular polygon are listed below.

Area of Regular Polygon

The region that the regular polygon occupies is known as its area. A polygon is classified as a triangle, quadrilateral, pentagon, etc. based on how many sides it has. The area of regular polygon is determined by:

Area of Regular Polygon (A) = [l2n]/[4tan(π/n)] units²

where,

• l is the side length
• n is the number of sides

Example: Determine the area of a polygon with 5 sides and a side length of 5 centimeters.

Solution:

Given,

• n = 5 cm
• l = 5 cm

Method for determining the region is,

A = [l2n]/[4tan(π/n)]

A = [5² x 5] / [4 tan(180/5)]

A = 125 / 4 x 0.7265

A = 43.014 cm²

Thus, the area of the polygon with five(5) sides is 43.014 cm²

Regular Polygon Perimeter Formula

The Perimeter of an n-sides regular polygon is can be calculated using the formula.

Perimeter (P) = n × s

where,

• "n" represents the number of sides of a regular polygon
• "s" represents the length of the side of Regular Polygon

Example: Find the perimeter of the hexagon with a length of 7 cm.

Solution:

Given,

Length of Side = 7 cm

For Hexagon,

n = 6

Perimeter of Regular Polygon(P) = n × s

P = 6 × 7 = 42 cm

Thus, the perimeter of the hexagon is 42 cm

Irregular Polygon Vs Regular Polygon

Regular polygons provide symmetrical forms and consistency which makes them simple to recognize. Contrarily, irregular polygons are less predictable and more difficult to categorize since their sides and angles have various lengths and measurements, resulting in asymmetrical forms.

The major differences between a Regular polygon and an Irregular polygon are discussed in the table below,

Read More,

• Polygon Formula
• Types of Polygons
• Area of Regular Polygon

Solved Examples on Regular Polygons

Example 1: If a polygon has 40 external angles, then the number of sides it has,

Solution:

Given

• Each Exterior Angle(n) = 40°

Number of Sides = 360°/n [Formula of regular polygon on the Exterior Angles]

Number of Sides = 360°/40° = 9

Thus, the number of sides in the polygon is 9

Example 2: Calculate the number of diagonals in a regular polygon with 24 sides.

Solution:

Given,

Number of sides in regular polygon = 24 sides

Formula for diagonals of the regular polygon

Number of diagonals in n sides polygon(N) = n(n - 3)/2

N = 24(24-3)/2 = 252

Thus, the number of diagonals in a polygon with 24 sides is 252

Example 3: What is the number of sides of a regular polygon if each interior angle is 90°?

Solution:

Given,

Each Interior Angle = 90°

Formula of interior angle of an n-sided regular polygon,

Each Interior Angle = [(n - 2) x 180°]/n

90° = [(n - 2) x 180°]/n

90n = (n - 2) x 180°

90n = 180n - 360

90n-180n = - 360

-90n = -360

n = 4

Thus, the number of sides in the regular polygon with 90 degree interior angle is 4

Conclusion

Regular polygons, with their equal sides and angles, are a fundamental concept of geometric. Their symmetry and uniformity not only make them unique but also incredibly useful in various applications, from architecture to everyday design. Their predictable and uniform structure makes them ideal for creating patterns, tiling surfaces, and designing various objects that require both form and function.