Regular and Irregular Polygons

Based on their sides, all polygons are broadly classified into two groups: regular and irregular. Here we will discuss them one after the other.

Regular Polygon

A regular polygon is a two-dimensional shape having all sides of equal length and all interior angles of equal measure. Thus sides and angles are the two parts of a regular polygon that are always congruent. They are thus both equilateral and equiangular. A regular polygon can be both convex and concave.

How Many Sides Does a Regular Polygon Have (With Examples)

As we know, a polygon is any shape having three or more sides, a regular polygon can be made of any number of sides.

The common regular polygons with their characteristic features are given below:

Formulas

Interior Angle

Interior angles are the angles that are formed inside the polygon at its corners when the line segments join in an end to end fashion. Here, we will discuss about how to find the measure of the sum of the interior angles in any n-sided regular polygon and also how to find the individual angles.

Sum of Interior Angles

It is the total measure of all the interior angles combined in the polygon. To determine the sum of the interior angles in a regular polygon, we divide the polygon into triangles. Since the sum of interior angles in a triangle is 180Â°, multiplying the number of triangles in the polygon by 180Â° will give us the sum of the interior angles in a regular polygon.

The standard formula to determine the sum of all angles in any regular polygon is given below:

Calculate the sum of the interior angles in a 22-sided regular polygon.

Solution:

As we know,
Sum of the measure of interior angles = (n-2) Ã— 180Â°, here n = 22
Â = (22-2) x 180Â°
= 3600Â°

One Interior Angle

Since in a regular polygon all the sides are equal, knowing the sum of all the interior angles we can easily determine the measure of any one angle by simply dividing the sum with the number of sides present in the polygon. The formula to determine the measure of each angle in a regular polygon is given below:

Find the measure of the interior angles of a regular polygon having 12 sides.

Solution:

As we know,
Interior angle of a Regular Polygon = (n-2) Ã— 180Â° /n, here n = 12
Â = (12-2) x 180Â° /12
= 10 x 180Â° /12
= 150Â°

Given below is a table showing the measure of the interior angles and their sums of some commonly studied regular polygons.

Exterior Angle

Exterior angle is the angle formed by any side of the polygon and the extension of its adjacent side.

How to Find the Exterior Angle of a Regular Polygon

For a regular polygon, all the exterior angles are equal, and they add up to 360Â°. Thus dividing 360Â° by the number of sides will give us the measure of each exterior angle of the regular polygon. The formula for calculating the size of an exterior angle is given below:

Calculate the exterior angle in a 10-sided regular polygon.

Solution:

Â As we know,
Exterior angle of a Regular Polygon = 360Â°/n, here n =10
Â = 360Â°/10
= 36Â°

Central Angle

The central angle of a polygon is the angle formed at the center of the polygon by any two adjacent vertices. The number of central angles in a polygon is equal to the number of sides of the polygon. Since a regular polygon has all equal sides, the central angles are also equal.

How to Find the Central Angle of a Regular Polygon

Since all the central angles of the polygon together form a complete circle, thus the sum of all the central angles of a polygon add up to 360Â°. The formula for finding the central angles of a polygon is thus same as the formula used to determine the exterior angle, which is given below:

Calculate the central angles in a 20-sided regular polygon.

Solution:

Â As we know,
Central angles of a Regular Polygon = 360Â°/n, here n =20
Â = 360Â°/20
= 18Â°

Finding the number of sides of sides of the regular polygon when only CENTRAL ANGLES are known

Find the number of sides in a regular polygon with each central angle measuring 72Â°

Solution:

As we know,
Central angles of a Regular Polygon = 360Â°/n, here central angle = 72Â°
=> 72Â° = 360Â°/n
=> n = 360Â°/72Â°
=> n = 5

Apothem

The apothem of a polygon is the line segment joining the center of the polygon to the midpoint of one of its sides.

How to Find the Apothem of a Regular Polygon

The apothem of a regular polygon can be determined when both the area and perimeter of the polygon are known. To determine the apothem, we will use the formula for finding the area of the polygon, as shown below:

As we know,

Area (A) = Â½ x p x a, here p = perimeter, a = apothem

Now, multiplying by 2 and dividing by p we get,

2A/p = a

a = 2A/p

Find the apothem of a regular n-sided polygon having an area of 96.44 square units and perimeter of 35 units.

Solution:

As we know,
Apothem (a) = 2A/p, here a = 96.44, p = 35
Â = 2 x 96.44/35
= 5.510 units

The radius of a polygon is the distance from the center of the polygon to any vertex. The formula to determine the radius is given below:

Find the radius of a six-sided regular polygon with side length of 12 cm.

Solution:

Radius (r) = s/2sin(180/n), here s = 12 cm, n = 6
= 12/2 x sin(180/6)
Â = 6 x sin 30
= 6 x 0.5
= 3 cm

Finding the radius of the regular polygon when NUMBER OF SIDES and APOTHEM are known.

Find the radius of a six-sided regular polygon with apothem length of 8 cm.

Solution:

Radius (r) = a/cos(180/n), here a = 8 cm, n = 6
= 8/cos(180/6)
= 8/cos30
= 8/âˆš3/2
= 8/0.86
= 9.3 cm

Irregular Polygon

An irregular polygon, also known as non-regular polygon is a shape that does not have all sides of equal length and all angles of equal measure. Thus any polygon that is not regular is an irregular polygon.

Examples

A scalene triangle, rectangle, trapezoid or a kite are common examples of irregular polygons. Pentagons and hexagons can also be irregular polygons when they do not have equal sides and angles.

FAQs

Q1. Is a triangle a regular polygon?

Ans. A triangle being the simplest polygon with three sides they can be both regular and irregular. For example, an equilateral triangle is a regular polygon whereas isosceles or scalene triangles are irregular polygons.

Q2. Is a rectangle a regular polygon?

Ans. No, a rectangle is not a regular polygon because all its sides are not of equal length.

Q3. Is a hexagon a regular polygon?

Ans. A hexagon with all equal sides and angles are regular polygons whereas an irregular hexagon with one or more sides and angles unequal are irregular polygons.

Q4. Is a rhombus a regular polygon?
Ans. No, a rhombus is not a regular polygon because all its angles are not of equal measure.

Q5. Is a pentagon a regular polygon?

Ans. A pentagon with all equal sides and angles are regular polygons whereas an irregular pentagon with one or more sides and angles unequal are irregular polygons.

Q6. Is a square a regular polygon?

Ans. Yes, a square is a regular polygon because all its sides are of equal length and all its angles are of equal measure.

Q7. Is a trapezoid a regular polygon?

Ans. No, a trapezoid or a trapezium is not a regular polygon because all its sides are not of equal length and all angles are not of equal measure.

Q8. Is a scalene triangle a regular polygon?

Ans. No, a scalene triangle is not a regular polygon because all its three sides are of unequal length.

Q9. What two parts of a regular polygon are congruent?

Ans. Two parts of a regular polygon that are congruent are sides and angles.