Irregular Polygons are closed shapes which we often come across in the study of Plane Geometry. While regular polygons are always convex, irregular polygons can be a convex or a concave Polygon. Even though it appears that regular Polygons were perfect in shape, in real life we generally come across only irregular Polygons. Fields, Yards, Housing plots and many interior spaces are found to be of irregular polygon shapes. And it is often required to find the measures related to these shapes. This makes the study of irregular polygons necessary in Geometry.

The fact that a polygon can be partitioned into smaller polygons helps us in dividing an irregular polygons into a number of regular and irregular polygons. As the areas of these smaller polygons can be calculated using formulas, the area of an irregular polygon can also be arrived as a sum.

A Regular Polygon is defined to be a convex polygon where all the sides are congruent and all the angles or congruent.

Hence an irregular polygon is a polygon which is not regular. That is either all sides are not congruent or all angles are not congruent or both the conditions for regular polygon fail. Regular Polygons can be either convex or concave.

A Polygon is termed to be irregular if:

All sides are not congruent
All angles are not congruent
both the sides are not congruent and the angles are not congruent.

Let us look into few examples of irregular polygons.
Some commonly known and well defined plane figures are irregular polygons.
  • Scalene and isosceles triangles are Irregular Polygons.

  • A rectangle is equiangular, but not equilateral. Hence rectangles are Irregular Polygons.

  • Parallelograms and Rhombuses are irregular Polygons. A Rhombus is equilateral but not equiangular.
Parallelogram Rhombus

  • Trapezoids are neither equilateral nor equiangular hence they are irregular Polygons.

  • A Kite is an irregular Polygon

The above shapes are convex irregular Polygon.
  • The Star like shape given below is a concave irregular Polygon.

Even though star shaped Dodecagon (12 sided Polygon) is equilateral it is not equiangular.
If an irregular Polygon of n sides is equilateral, then its Perimeter is given by
P = n.s units

where s is the measure of the side of the Polygon.

Example: The perimeter of a Rhombus with side measure = 6 cm is
P = 4 x 6 = 24 cm.

If the Polygon is not equilateral, then the Perimeter is got by adding up the measures of all sides.

Example: Find the perimeter of the irregular Polygon shown below:

The perimeter is got by adding the measures of all the sides. The figure is not done to scale.
P = 4 + 3 + 7.5 + 5 + 3.5 = 23 meters.
Areas of irregular Polygons like triangles, rectangles and other quadrilaterals can be found using the corresponding formulas.

Solved Examples

Question 1: Find the area of a parallelogram with a side measuring 10 inches and the length of the corresponding altitude = 8 inches.
Area of a Parallelogram = Length of side x Length of the altitude
                                      = 10 x 8 = 80 sq. inches.

Other complex shapes can be partitioned into smaller Polygons whose areas can be calculated using formulas. Then the area of the larger Polygon is got by summing up the areas so found.

Question 2: A Layout is given below. Find the total area.

The irregular Polygon can be partitioned into three rectangles as follows:

Area Partition

Area of a rectangle = length x width
Area of rectangle 1 = 80 x 60 = 4,800 sq.meters
Area of rectangle 2 = 70 x 20 = 1,400 sq.meters.
Area of rectangle 3 = 50 x 30 = 1,500 sq.meters

Hence the area of the Layout = 4,800 + 1,400 + 1,500 = 7,700 sq.meters.

Let us compare the characteristics of regular and irregular Polygons.

Regular Polygons
Irregular Polygons
All Regular Polygons are equilateral
Some irregular Polygons are equilateral.
All Regular Polygons are equiangular
Some irregular Polygons are equiangular.
All Regular Polygons are both equilateral
and equiangular
No irregular Polygon is both equiangular
and equilateral.
Regular Polygons are convex polygons.
Irregular Polygons are convex or concave.
Sum of interior angles = (n - 2) 180 degrees
Sum of interior angles = (n - 2) 180 degrees
if the Polygon is convex.
Measure of an exterior angle
= $\frac{360}{n}$ degrees
Measure of each exterior angle varies.
Area of a regular Polygon
= $\frac{1}{2}$ x Perimeter x Apothem
No single formula exists to compute area.