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**Irregular Polygon|Definition & Meaning**

**Definition**

A **polygon** is **considered** **irregular** if **neither** its **sides** nor its **angles** are of the **same** **length**. It **differs** from a **regular** **polygon**, which is a **polygon** with **all** **sides** being the **same** **length** and all angles being the same size. Since **irregular** **polygons** **lack** the **symmetry** and **regularity** of regular polygons, they can be difficult to work with in geometry.

**Overview**

Figure 1 – A simple irregular Polygon

A crucial **characteristic** of an **irregular** **polygon** is that it is **not** **cyclic**, which **prevents** it **from** being **encircled** **by** a **circle**. This is due to the fact that an **irregular** **polygon’s** **sides** and **angles** are **not** **equal**, making it **impossible** to **distribute** them **evenly** around a **focal** **point**. However, **irregular** **polygons** can be **circumscribed** **around** a **circle**, which **means** that the **polygon’s** **perimeter** can be **enclosed** by a **circle**. The above figure shows an irregular polygon.

**Properties of Irregular Polygon**

There are several properties that are unique to irregular polygons:

**Irregular polygons cannot**be**encircled**by a circle because, as was already mentioned, they are**not cyclic**. An irregular polygon**cannot**be**dispersed uniformly**around a central point because its sides and angles are not equal.**Self-Intersecting:**Some asymmetric polygons have**sides**that**cross one another**to**form**a**shape**that is**self-intersecting.**The**bowtie form**is an**illustration**of a**polygon**that**self-intersects.****Star polygons**are**irregular polygons**that have**points**or**vertices**that**radiate outward**from the center of the pattern, and they can also be**categorized**as**irregular polygons.**The**pentagram**is an**illustration**of a**star polygon**that began by**cutting**the**polygon**into**smaller,**more**recognizable forms,**and calculating the areas of each.**Non-convex:**Some irregular polygons fall into this category if**at least****one**of their**angles exceeds 180 degrees.**The polygon is**“bent”**or**“curved” inwards**in this sense.**Complex Shapes:**Unlike regular polygons, irregular polygons can**take on**a**number**of**complex shapes,**giving them a**wider range**of possible**uses.****Internal Angles:**A formula that takes into account the polygon’s number of sides can be used to**compute**the**sum**of the**interior angles****of**an**irregular polygon.**To get the size of each angle in an irregular polygon, apply this formula.**Perimeter:**An irregular polygon’s**perimeter**is**equal**to the**sum**of all of its**sides.****Area:**An**irregular polygon’s area**can be**measured**by**breaking**it up**into smaller,**more recognizable shapes, then calculating the areas of each of those shapes.**Lines connecting**the**non-adjacent vertices**of an irregular polygon are**known**as its**diagonals.**A formula that considers the number of sides in the polygon can be used to determine the number of diagonals in an irregular polygon.

**Regular vs. Irregular Polygon**

**Regular Polygon**

A **regular polygon** is one that **has sides** that are the **same length** and **angles** that are the **same size.** In other terms, a regular polygon has **congruent sides** and **angles. Equilateral triangles, squares, pentagons, hexagons,** and **octagons** are some examples of regular polygons.

**Irregular** **Polygon**

An **irregular polygon** is one that has sides that **vary** in **length** and **angles** that vary in **size.** In other terms, an irregular polygon **does not** have **congruent sides** and angles. **Scalene triangles, rectangles,** and **heptagons** are a few examples of irregular polygons.

**Difference**

By **examining** their **symmetry, regular** and **irregular polygons** can be **distinguished** from one **another.** A **regular polygon** is **symmetrical,** which means that each half would be a mirror image of the other if a line of symmetry were drawn through its center. On the other hand, an **irregular polygon lacks** this **symmetry.**

**Types of Irregular Polygons**

**Scalene Triangle**

An irregular polygon, more precisely a **triangle,** with **no sides** of **equal length** is called a **scalene triangle.** It **doesn’t** have **equal-length** sides, so it doesn’t have **equal-sized angles** either. Inferred from the fact that a scalene triangle **lacks equal-length** sides and angles, it follows that a scalene triangle is an **irregular polygon.** The irregular polygon is shown in the below figure.

Figure 2 – Scalene Triangle an irregular Polygon

**Depending** on **how big** the angles are, a scalene triangle can be acute, right, or obtuse. All of the **angles** in a **scalene triangle** are **fewer** than **90** degrees in an acute triangle, 90 degrees in a **right scalene** triangle, and **greater** than **90** degrees but **less than 180** degrees in an **obtuse scalene** triangle.

**Right Angle Triangle**

A **right triangle** is a particular kind of irregular polygon that has **one angle** that is **exactly 90 degrees.** The right angle is the name for this angle. It is regarded as an **irregular polygon** because **one** of its **angles isn’t** the **same** as the rest. The irregular polygon is shown in the below figure.

Figure 3 – Right Angled Triangle is an irregular polygon.

A right triangle has **two sides unequal** in **length,** with **one side** being **longer** than the **other,** and **one right angle.** The triangle’s two shorter sides are referred to as the triangle’s legs, while its longer side is known as the hypotenuse. The **legs** are the **two** **sides** that are **next** to the **right angle,** and the **hypotenuse** is the **side** that is **opposite** the **right angle.**

**Isosceles Triangle**

An **isosceles triangle** is a particular kind of triangle in **which** the **lengths** of the **two sides** are **equal.** Its **uneven shape** is due to the fact that not all of its sides are the same length.

The **legs** and **base** of an **isosceles triangle** are its **two equally long** sides. The **third side** is referred to as the **hypotenuse.** The **vertex** is the **intersection** of the two legs, and the angle between them is referred to as the vertex angle. The irregular polygon is shown in the below figure.

Figure 4 – Isosceles Triangle is an irregular polygon.

Depending on how long the legs are in relation to how long the base is, **isosceles triangles** can have **two acute angles or** two **obtuse** angles. The **triangle** has two **acute angles if** the **legs** are **shorter** than the **base.** If the **legs** are **longer** than the **base,** the **triangle** will have two **obtuse angles.**

**Rectangle**

**Four-sided polygons** with **opposite sides** that are **parallel** and **equal** in length make up rectangles. Its uneven shape is due to the fact that not all of its sides are the same length.

Figure 5 – Rectangle an irregular polygon.

The **parallel, equal-length** sides of a **rectangle** are **referred** to as the **length** and **width.** Usually, the **length** is **more** than the **width.** The width is **represented** by the two shorter sides and the length by the two longer sides. A **rectangle’s** area is equal to the **product** of its **length** and **width,** and its **perimeter refers** to the **distance around** its **outside.** The irregular polygon is shown in the above figure.

**Example Involving Irregular Polygon Calculations**

Figure 6 – Area of Irregular Polygon example

**Consider** an **irregular polygon** shown in the figure. **Find** the **area** of this polygon.

**Solution**

As this is a **rectangle** so we can **find** its **area** by the **formula** of the area of a rectangle.

Area of rectangle = Length x Width

Area of rectangle = 6 x 4

Area of rectangle = 12 sq. units

*All mathematical drawings and images were created with GeoGebra.*