Irregular Polygons
Irregular polygons are those types of polygons that do not have equal sides and equal angles. In other words, irregular polygons are not regular. Polygons are closed two-dimensional figures that are formed by joining three or more line segments with each other. There are two types of polygons, regular and irregular polygons. Let us learn more about irregular polygons, the types of irregular polygons, and solve a few examples for better understanding.
Definition of Irregular Polygons
Irregular polygons are shapes that do not have their sides equal in length and the angles equal in measure. Hence, they are also called non-regular polygons. We experience irregular polygons in our daily life just as how we see regular polygons around us. The shape of an irregular polygon might not be perfect like regular polygons but they are closed figures with different lengths of sides. Some of the examples of irregular polygons are scalene triangle, rectangle, kite, etc. When the angles and sides of a pentagon and hexagon are not equal, these two shapes are considered irregular polygons. The image below shows some of the examples of irregular polygons.
Properties of Irregular Polygons
Irregular polygons have a few properties of their own that distinguish the shape from the other polygons. The properties are:
- An irregular polygon does not have equal sides and angles.
- Irregular polygons can either be convex or concave in nature.
- Irregular polygons are shaped in a simple and complex way.
- Irregular polygons are infinitely large in size since their sides are not equal in length.
- Shapes like parallelograms, trapeziums, and quadrilaterals are considered irregular polygons as their adjacent sides and adjacent angles are not equal.
Types of Irregular Polygons
There are different types of irregular polygons. However, we are going to see a few irregular polygons that are commonly used and known to us. Let's take a look.
Scalene Triangle
A scalene triangle is considered an irregular polygon, as the three sides are not of equal length and all the three internal angles are also not in equal measure and the sum is equal to 180°. In the triangle PQR, the sides PQ, QR, and RP are not equal to each other i.e. PQ ≠ QR ≠ RP. Also, angles ∠P, ∠Q, and ∠R, are not equal, ∠P ≠ ∠Q ≠ ∠R. Thus, we can use the angle sum property to find each interior angle.
Isosceles Triangle
An isosceles triangle is considered to be irregular since all three sides are not equal but only 2 sides are equal. All three angles are not equal but the angles opposite to equal sides are equal to measure and the sum of the internal angles is 180°. In the triangle, ABC, AB = AC, and ∠B = ∠C. All the three sides and three angles are not equal.
Rectangle
A rectangle is considered an irregular polygon since only its opposite sides are equal in equal and all the internal angles are equal to 90°. In the given rectangle ABCD, the sides AB and CD are equal, and BC and AD are equal, AB = CD & BC = AD. And, ∠A = ∠B = ∠C = ∠D = 90 degrees. But,
AB ≠ AD or BC
BC ≠ AB or CD
CD ≠ AD or BC
AD ≠ AB or CD
Hence, the rectangle is an irregular polygon.
Right Triangle
A right triangle is considered an irregular polygon as it has one angle equal to 90° and the side opposite to the angle is always the longest side. Therefore, the lengths of all three sides are not equal and the three angles are not of the same measure. In the right triangle ABC, the sides AB, BC, and AC are not equal to each other. AB = BC = AC, where AC > AB & AC > BC. And, ∠x ≠ ∠y ≠ ∠z, where ∠y = 90°.
Irregular Pentagon
A pentagon is considered to be irregular when all five sides are not equal in length. However, sometimes two or three sides of a pentagon might have equal sides but it is still considered as irregular.
Irregular Hexagon
A hexagon is considered to be irregular when the six sides of the hexagons are not in equal length. The measurement of each of the internal angles is not equal. By the below figure of hexagon ABCDEF, the opposite sides are equal but not all the sides AB, BC, CD, DE, EF, and AF are equal to each other. Since the sides are not equal thus, the angles will also not be equal to each other. Therefore, an irregular hexagon is an irregular polygon.
Difference Between Irregular and Regular Polygons
A polygon can be categorized as a regular and irregular polygon based on the length of its sides. As the name suggests regular polygon literally means a definite pattern that appears in the regular polygon while on the other hand irregular polygon means there is an irregularity that appears in a polygon. Let us see the difference between both.
| Regular Polygons | Irregular Polygons |
| The length of the sides of a regular polygon is equal. | The length of the sides of an irregular polygon is not equal. |
| The measurement of all interior angles is equal. | The measurement of all interior angles is not equal. |
| The measurement of all exterior angles is equal. | The measurement of all exterior angles is not equal. |
| A polygon that is equiangular and equilateral is called a regular polygon. | A polygon whose sides are not equiangular and equilateral is called an irregular polygon. |
Irregular Polygons Formulas
Calculating the area and perimeter of irregular polygons can be done by using simple formulas just as how regular polygons are calculated. Let us look at the formulas:
Area of Irregular Polygons
An irregular polygon is a plane closed shape that does not have equal sides and equal angles. Thus, in order to calculate the area of irregular polygons, we split the irregular polygon into a set of regular polygons such that the formulas for their areas are known. Consider the example given below.
The polygon ABCD is an irregular polygon. Thus, we can divide the polygon ABCD into two triangles ABC and ADC. The area of the triangle can be obtained by:
Area of polygon ABCD = Area of triangle ABC + Area of triangle ADC.
Perimeter of Irregular Polygons
Polygons that do not have equal sides and equal angles are referred to as irregular polygons. Thus, in order to calculate the perimeter of irregular polygons, we add the lengths of all sides of the polygon.
Example: Find the perimeter of the given polygon.
Solution: As we can see, the given polygon is an irregular polygon as the length of each side is different (AB = 7 units, BC = 8 units, CD = 3 units, and AD = 5 units)
Thus, the perimeter of the irregular polygon will be given as the sum of the lengths of all sides of its sides.
Thus, the perimeter of ABCD = AB + BC + CD + AD ⇒ Perimeter of ABCD = (7 + 8 + 3 + 5) units = 23 units
Therefore, the perimeter of ABCD is 23 units.
Sum of Interior Angles of Irregular Polygons
The interior angles of a polygon are those angles that lie inside the polygon. Observe the interior angles A, B, and C in the following triangle. The interior angles in an irregular polygon are not equal to each other. Therefore, to find the sum of the interior angles of an irregular polygon, we use the formula the same formula as used for regular polygons. The formula is: Sum of interior angles = (n − 2) × 180° where 'n' = the number of sides of a polygon.
Example: What is the sum of the interior angles in a Hexagon?
Solution:
A hexagon has 6 sides, therefore, n = 6
The sum of interior angles of a regular polygon, S = (n − 2) × 180
S = (6-2) × 180°
⇒ S = 4 × 180
⇒ S=720°
Therefore, the sum of interior angles of a hexagon is 720°.
Sum of Exterior Angles in Irregular Polygons
An exterior angle (outside angle) of any shape is the angle formed by one side and the extension of the adjacent side of that polygon. Observe the exterior angles shown in the following polygon.
To calculate the exterior angles of an irregular polygon we use similar steps and formulas as for regular polygons. The sum of the exterior angles of a polygon is equal to 360°. Therefore, the formula is,
Sum of exterior angles = 180n – 180(n-2) = 180n – 180n + 360. Hence, the sum of exterior angles of a pentagon equals 360°.
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Examples on Irregular Polygons
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Example 2: Find the area of the polygon given in the image.
Solution: It can be seen that the given polygon is an irregular polygon. The area of polygon can be found by dividing the given polygon into a trapezium and a triangle where ABCE forms a trapezium while ECD forms a triangle. In order to find the area of polygon let us first list the given values:
For trapezium ABCE,
Length of AB = 4 units
Length of EC = 7 units
Height of the trapezium = 3 units
Thus, the area of the trapezium ABCE = (1/2) × (sum of lengths of bases) × height = (1/2) × (4 + 7) × 3
⇒ Area of trapezium ABCE = (1/2) × 11 × 3 = 16.5 square unitsFor triangle ECD,
Length of EC = 7 units
Height of triangle = (6 - 3) units = 3 units
Thus, the area of triangle ECD = (1/2) × base × height = (1/2) × 7 × 3
⇒ Area of triangle ECD = (1/2) × 7 × 3 = 10.5 square unitsThe area of the polygon ABCDE = Area of trapezium ABCE + Area of triangle ECD = (16.5 + 10.5) square units = 27 square units
Therefore, the area of the given polygon is 27 square units.
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Example 3: Find the missing length of the polygon given in the image if the perimeter of the polygon is 18.5 units.
Solution: It can be seen that the given polygon is an irregular polygon. The perimeter of the given polygon is 18.5 units. The given lengths of the sides of polygon are AB = 3 units, BC = 4 units, CD = 6 units, DE = 2 units, EF = 1.5 units and FA = x units.
Given that, the perimeter of the polygon ABCDEF = 18.5 units
⇒ Perimeter of polygon ABCDEF = AB + BC + CD + DE + EF + FA = 18.5 units ⇒ (3 + 4 + 6 + 2 + 1.5 + x) units = 18.5 units. Thus, x = 18.5 - (3 + 4 + 6 + 2 + 1.5) = 2 unitsTherefore, the missing length of polygon ABCDEF is 2 units.
FAQs on Irregular Polygons
What are Irregular Polygons?
Irregular polygons are the kinds of closed shapes that do not have the side length equal to each other and the angles equal in measure to each other. In other words, irregular polygons are non-regular polygons. Due to the sides and angles, some convex and concave polygons can also be considered as irregular.
How to Find the Perimeter of Irregular Polygons?
In order to calculate the value of the perimeter of an irregular polygon we follow the below steps:
- Find the measurement of each side of the given polygon (if not given).
- Once the lengths of all sides are obtained, the perimeter is found by adding all the sides individually.
How Do You Find the Measure of an Interior Angle of an Irregular Polygon?
The measure of an interior angle of an irregular polygon is calculated with the help of the formula: 180° × (n-2)/n, where 'n' is the number of sides of a polygon.
How Do You Find the Measure of an Exterior Angle of an Irregular Polygon?
The measure of an exterior angle of an irregular polygon is calculated with the help of the formula: 360°/n where 'n' is the number of sides of a polygon.
How to Find the Area of Irregular Polygons?
In order to calculate the value of the area of an irregular polygon we use the following steps:
- Divide the given polygon into smaller sections forming different regular or known polygons.
- Find the area of each section individually.
- Add the area of each section to obtain the area of the given irregular polygon.