Regular Polygon

Regular polygons are closed two-dimensional planar figures constructed entirely of straight lines. In contrast to a regular polygon, which is made up of solely straight lines of equal length, an irregular polygon has varied sides and angles. Polygons are two-dimensional geometric shapes with a fixed number of sides.

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 its sides are congruent.

Regular Polygon Definition

Regular polygons have the same number of sides and interior angles. Squares, rhombuses, equilateral triangles, and other shapes serve as cases of regular polygons. Regular polygons have both congruent angles and congruent sides. They are equiangular, according to that.

Regular Polygons Examples

In real life, regular polygons are abundant in our surroundings. A common example is a stop sign, an octagon with eight equal sides. Road signs often employ triangles, rectangles, 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.

Regular Polygon Shape

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 as follows:

Equilateral Triangle

The regular polygon with the smallest possible number of sides is Equilateral Triangle.

Properties of Equilateral Triangle

Various properties of the equilateral triangle are,

• The sum of the interior angle of an equilateral triangle is 180°.
• It has 3 equal sides.
• It’s each interior angle is 60° and each exterior angle is 120°.
• There are 0 diagonals in the equilateral triangle.
• Perimeter of the equilateral triangle is 3 x Side.

Square

A 4-sided regular polygon, or square, is a quadrilateral with four equal sides.

Properties of Square

Various properties of the square are,

• The sum of the interior angle of the square is 360°.
• It has 4 equal sides.
• It’s each interior angle is 90° and each exterior angle is 90°.
• There are 2 diagonals and 2 triangles formed.
• Perimeter of square is 4 × Side.

Regular Pentagon

A regular Pentagon is a Pentagon or 5-gon with equal sides that are shown in the following diagram:

Properties of Regular Pentagon

Various properties of the regular pentagon are,

• The sum of the interior angle is 540°.
• It has 5 equal sides.
• It’s each interior angle is 108° and each exterior angle is 72°.
• There are 5 diagonals and 3 triangles formed.
• Its Perimeter is 5 x side.

Regular Hexagon

Hexagons are six-sided polygons, and when all of their sides are equal, they are referred to as regular hexagons.

Properties Regular Hexagon

Various properties of the regular hexagon are,

• The sum of the interior angle is 720°.
• It has 6 equal sides.
• It’s each interior angle is 120° and each exterior angle is 60°.
• There are 9 diagonals and 4 triangles formed.
• Its Perimeter is 6 x side.

Regular Heptagon

A Heptagon is a seven-sided polygon with equal sides and angles, they are referred to as regular heptagons.

Properties of Regular Heptagon

Various properties of the regular heptagon are,

• The sum of the interior angle is 900°.
• It has 7 equal sides.
• Each interior angle is 128.57° and each exterior angle is 51.428°.
• There are 14 diagonals and 5 triangles formed.
• Its Perimeter is 7 x side.

Regular Octagon

An eight-sided polygon with equal sides and angles is called a regular octagon.

Properties of Regular Octagon

Various properties of the regular octagon are,

• The sum of the interior angle is 1080°.
• It has 8 equal sides.
• It’s each interior angle is 135° and each exterior angle is 45°.
• There are 20 diagonals and 6 triangles formed.
• Its Perimeter is 8 x side.

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 of 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

• If the sides of a regular polygon are n, then the number of triangles generated by connecting the diagonals from one of the polygon’s corners is n – 2.

Formulas for Regular Polygons

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.

Regular Polygon Area Formula

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 regular polygon’s area 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

How to Calculate Area of a Regular Polygon?

The area of a regular polygon is calculated using the steps discussed below,

Step 1: Mark the length of each side and the number of the sides of given regular polygon.

Step 2: Use the area of the regular polygon formula discussed above with the values from Step 1.

Step 3: Simplify the values from Step 2 to find the required area of the given polygon.

Regular Polygon Perimeter Formula

A regular polygon’s perimeter may be calculated using a simple formula. The perimeter (P) of a regular polygon with n sides and s sides may be computed by applying the formula below:

Perimeter (P) = n × s

where,

• n is number of Sides in a Regular Polygon
• s is 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 like squares, triangles, and pentagons because their sides and angles are equal. Their consistency 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 image added below shows the regular polygon and the irregular polygon.

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

Irregular Polygon	Regular Polygon
Lengths of the sides vary in the irregular polygon	Each side is of the same length for the regular polygon.
Interior angles in the irregular polygon are different.	Each internal angles in the regular polygon are the same.
Examples include irregular forms that are not consistent.	Squares, Triangles, Pentagons, Hexagons, and other shapes with equal sides are examples of regular polygons.
Identification and classification might be difficult due to their diverse features.	Simple to classify and identify.

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

FAQs on Regular Polygons

What is a Regular Polygon?

A regular polygon is one that has equal sides and angles. It is identified by having an equal number of sides and equal measurements for each of its inner angles. Equilateral triangles, squares, and hexagons are examples of regular polygonal forms.

How Many Sides a Regular Polygon has?

A regular polygon is one with an equal number of sides and angles. However, it must be a positive integer bigger than or equal to three, regardless of the number of sides. Examples include triangles with three sides, squares with four sides, pentagons with five sides, hexagons with six sides, etc.

What are Properties of Regular Polygons?

The major properties of the regular polygon are,

• A regular polygon has an equal number of sides.
• Interior angles are the regular polygon all equal.
• A regular polygon’s perimeter is equal to the side measure multiplied by n for every n side.
• Exterior angles of the regular add up to 360°. (In fact, exterior angles of any polygon add up to 360°)

How to Calculate Area and Perimeter of a Regular Polygon?

The formulas used to calculate the area and perimeter of the regular polygon are,

Area (A) = [l²n] / [4tan(π/n)] units²

And

Perimeter (P) = n × s units

where,

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

What are Examples of Regular Polygon?

Regular polygons may be found everywhere in daily goods and architecture. Typical windowpanes are square, stop signs are octagons, and ceiling tiles frequently resemble regular hexagons. Regular polygons are useful and aesthetically pleasing for a variety of designs and structures due to their regular forms.