Irregular Polygons – Definition, Types, Formula
11 minutes read
Created: December 19, 2023
Last updated: January 5, 2024
Geometry is an infinite playground of shapes, where every figure has a unique story to tell, and one such intriguing protagonist is the irregular polygon. Often overshadowed by their wellstructured siblings, the regular polygons, irregular polygons hold a charm of their own. At Brighterly, we believe in exploring every nook and cranny of this mathematical wonderland, and today, we invite you on a journey into the world of irregular polygons.
These ‘rulebreakers’ of the polygon family stand apart due to their lack of uniformity. Each side and angle in an irregular polygon takes on a different length or degree, making them wonderfully unpredictable. But don’t be fooled! Even within their apparent chaos, they follow the unwavering rules of geometry, as we will uncover in the article.
What are Irregular Polygons?
Irregular polygons are fascinating shapes that play a prominent role in the intriguing world of geometry. They are a subcategory of polygons, which are closed shapes with straight sides. What sets irregular polygons apart is their lack of uniformity; unlike regular polygons, irregular polygons don’t have sides that are all the same length or angles that are all the same size. They break the mold, and it’s precisely this irregularity that makes them such a captivating area of study in geometry.
Kids, did you know that your favorite breakfast cereal might be an irregular polygon? Or that beautiful, cascading crystal chandelier in the living room? Yes, irregular polygons are all around us, adding a dash of geometric zest to our daily lives. So, let’s dive deeper into the intriguing world of irregular polygons.
Definition of Irregular Polygons
An irregular polygon is essentially a polygon that does not conform to the characteristics of a regular polygon. Specifically, it does not have all sides equal and all angles equal. This gives them their unique, often unpredictable shape. This may seem like they are ‘rulebreakers’, but even within their wild nature, they follow precise rules of geometry that we will explore in this article.
In simple words, the definition of an irregular polygon can be given as: a polygon with unequal sides and angles. For example, a square is not an irregular polygon because all of its sides and angles are the same. However, a rectangle is an irregular polygon because while all angles are equal, the sides are not.
Types of Irregular Polygons
Irregular polygons come in many different shapes and sizes. There are several types of irregular polygons based on the number of sides they have. Here are a few examples:

Irregular Quadrilateral: An irregular quadrilateral has four sides and four angles, but not all sides and angles are equal. An example is a scalene quadrilateral, where all sides and angles are different.

Irregular Pentagon: An irregular pentagon has five sides and five angles. However, not all sides and angles are equal. The shape of your typical home plate in a baseball field is a perfect example of an irregular pentagon.

Irregular Hexagon: An irregular hexagon has six sides and six angles, but not all of them are the same length or size.
Formula of Irregular Polygons
One of the most significant challenges when dealing with irregular polygons is finding their area and perimeter. Unlike their regular cousins, there is no onesizefitsall formula. The perimeter of an irregular polygon can be calculated by simply adding up the lengths of all the sides.
Finding the area, on the other hand, often requires you to divide the irregular polygon into shapes you know how to work with, like triangles and rectangles. For example, if you can split an irregular polygon into two triangles, you can calculate the area of each triangle using the formula 1/2 × base × height, and then add those areas together.
Properties of Irregular Polygons
Irregular polygons may appear unpredictable at first glance, but they have specific properties that give them a certain order. Let’s go through a few of these properties:

Sides: Irregular polygons have sides of varying lengths. There’s no restriction on how short or long each side can be compared to the others.

Angles: The internal angles in an irregular polygon can also vary. However, the sum of the internal angles will always adhere to the formula (n2) × 180°, where n is the number of sides.

Symmetry: Unlike regular polygons, irregular polygons do not always have lines of symmetry. That means you can’t always fold them in half and get two identical parts.

Diagonals: The number of diagonals in an irregular polygon can be calculated using the formula n(n3)/2, where n is the number of sides.

Perimeter and Area: As mentioned earlier, the perimeter is the sum of all sides, and the area often requires splitting the shape into smaller, regular shapes.
Differences Between Types of Irregular Polygons
Each type of irregular polygon is unique in its own right. The differences between types of irregular polygons primarily stem from the number of sides they possess, which directly influences their shape, angle sum, and number of diagonals. For instance, an irregular pentagon (five sides) will have a different overall shape and a higher sum of interior angles than an irregular quadrilateral (four sides).
The beauty of irregular polygons is in their diversity. Whether they have four, five, six, or more sides, each shape presents its own set of fascinating attributes and challenges.
Equations for Calculating Properties of Irregular Polygons
Calculating the properties of irregular polygons is an adventure in its own right. As mentioned before, the sum of interior angles of an irregular polygon can be found using the formula (n2) × 180°, where n is the number of sides. The number of diagonals can be calculated by the formula n(n3)/2.
However, there isn’t a universal formula to calculate the area of an irregular polygon, as it largely depends on the shape of the polygon. If it can be divided into regular shapes like rectangles or triangles, the areas of these shapes can be calculated and summed up.
Writing Equations for Irregular Polygons
Writing equations for irregular polygons might seem like a daunting task, but it’s a great exercise in logical thinking and problemsolving. Essentially, you’ll have to use the properties and formulas we’ve discussed to construct your equations. For example, if you know the lengths of the sides of an irregular polygon, you can write an equation to find the perimeter by adding them up.
Practice Problems on Irregular Polygons
Let’s put these ideas to practice with a few problems:

Find the perimeter of an irregular quadrilateral with side lengths of 3cm, 5cm, 7cm, and 9cm.

An irregular pentagon has interior angles of 100°, 120°, 110°, 130°, and 80°. Verify if the sum of these angles follows the formula for the sum of interior angles.
Conclusion
Our exploration of irregular polygons has taken us through some exhilarating twists and turns. We’ve decoded the definitions, delved into different types, and deciphered the formulas and properties of these quirky characters in the geometry universe. Irregular polygons, as we discovered, are a captivating blend of unpredictable shapes and steadfast mathematical rules.
Brighterly encourages every child to see the beauty in diversity and irregularity, and these polygons serve as perfect examples of that. The power of understanding and mastering irregular polygons lies not in memorization, but in grasping the rules that govern them and knowing when and how to apply these rules.
Remember, an irregular polygon is not just a shape; it’s a manifestation of the beautiful irregularities that make our world interesting. So the next time you draw an oddlooking figure or spot a funnyshaped cloud, know that you’ve met another irregular polygon, an irregular yet beautiful part of our mathematical world.
Frequently Asked Questions on Irregular Polygons
What is an irregular polygon?
An irregular polygon is a type of polygon that does not have all sides and angles equal. This means that the length of the sides and the degree of the angles can differ throughout the polygon. This differentiates them from regular polygons, where all sides and angles are equal.
How can we calculate the area of an irregular polygon?
Calculating the area of an irregular polygon often involves a bit of creativity and problemsolving. Because there isn’t a universal formula for all irregular polygons, we usually have to divide the irregular polygon into simpler shapes, like triangles and rectangles, whose area we know how to calculate. By adding up the areas of these smaller shapes, we can find the total area of the irregular polygon.
What are the differences between regular and irregular polygons?
The key difference between regular and irregular polygons lies in the uniformity of their sides and angles. In a regular polygon, all sides are of the same length and all angles are of the same degree. An equilateral triangle or a square are examples of regular polygons. In contrast, an irregular polygon does not have this uniformity. Each side can be of a different length, and each angle can be of a different degree. For example, a rectangle or a scalene triangle are irregular polygons.
Information Sources
 Irregular Polygons – Wikipedia
 Polygon – Britannica
 Polygon Basics – National Council of Teachers of Mathematics
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