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# Regular|Definition & Meaning

## Definition

In mathematics, **regular** means different things depending on which **branch** of mathematics we look at. Generally, it refers to something that is **well-defined**, often predictable, and easily **identifiable** because of unique patterns and characteristics. For example, a regular **shape** (in geometry) is defined as one where all the **sides** and **angles** are equal in length (as in a **square**).

**Figure 1** shows a **square** which is a **regular** polygon.

## Polygons

A polygon is a closed **two-dimensional** figure with no curved surfaces and is a plane or flat **shape** having different numbers of edges and vertices. **Figure 2** shows three different shapes of which only one is a **polygon**.

A **corner** or the point where two **edges** meet is known as the **vertex** of a polygon. A polygon consists of more than two **vertices**.

The two essential **features** of a polygon that define its **regularity** are its **sides** and **angles**.

### Sides

The **edges** of a polygon are also known as its **sides**. The sides are the **line** segments that make the polygon. **Different** polygons have different numbers of sides.

### Angles

The angles of the polygon are the **internal** angles formed between two **adjacent** sides inside the polygon. A polygon has a number of **angles** equal to the number of** vertices**.

**Figure 3** shows the **sides** and the internal **angles** of a triangle.

For a **regular** polygon, all the **sides** and internal **angles** must be **equal** to each other.

## Different Types of Regular Polygons

There are different **types** of regular **polygons** depending on the number of **sides**.

### Equilateral Triangle

A triangle consists of **three** sides and three internal angles. An equilateral triangle has all the **sides** of equal length and each angle **equal** to **60°**, making a sum of **180°** of all the internal **angles**.

### Regular Quadrilaterals

A quadrilateral consists of **four** sides and four angles. Different types of quadrilaterals depend on the **length** of the sides and the measure of their angles. A square, **rectangle**, and parallelograms are **quadrilaterals** with the sum of **360°** of all the internal angles.

A square is a **regular** quadrilateral with each **internal** angle equal to **90°**.

### Regular Pentagon

A pentagon consists of **five** sides, five vertices, and five internal angles. A **regular** pentagon has each internal **angle** equal to **108°**, with the sum of all angles equal to **540°**.

### Regular Hexagon

A regular hexagon has **six** internal **angles**, each measuring **120°**, and six **sides** of equal length.

### Regular Heptagon

A heptagon has **seven** equal sides and internal angles. A **regular** heptagon has an internal **angle** equal to **128.57°** and all sides of equal length. All the internal angles of regular heptagon add to **900°**.

### Regular Nonagon

A nonagon consists of **nine** sides and nine angles that sum to **1260°**. A regular nonagon has nine equal **sides** and angles, with each **internal** angle equal to **140°**.

### Regular Decagon

“Deca” means **ten**, so a decagon has ten **edges** and ten vertices. A regular decagon has all the sides of **equal** length, with each **interior** angle equal to **144°**. The **sum** of the internal angles of a decagon equals **1440°**.

**Figure 4** shows all types of **regular** polygons.

## Lines of Symmetry of Regular Polygons

The lines of **symmetry** in a regular **polygon** are imaginary lines that divide a regular polygon into **equal** parts and pass through its** center**. The lines of symmetry of a **regular** polygon equal its number of **sides**.

**Figure 5** shows the lines of symmetry of an **equilateral** triangle.

## Rotational Symmetry of Regular Polygons

The **rotational** symmetry of a **regular** polygon is a similar **view** of the polygon compared with the **original** view when rotated at some degrees other than **360°**. The **order** of a regular polygon’s rotational symmetry **equals** the number of its **sides**.

The **angle** at which the **polygon** is rotated is known as the angle of** rotation**. It can be calculated by dividing **360°** by the total number of **sides** of the polygon. For example, a **pentagon** is rotated at 360**°**/5 = **75°** to check its rotational symmetry.

## Irregular Polygons

The polygons having either **unequal** lengths of **sides** or unequal interior **angles** or both are known as **irregular** polygons. The following are some of the irregular polygons.

### Irregular Triangles

There are **three** main types of **irregular** triangles.

#### Isosceles Triangle

A triangle with **two** equal sides out of three sides is known as an **isosceles** triangle. The two interior angles **opposite** the two sides of **equal** length are also equal. All the sides and angles are not equal, which makes it an **irregular** polygon.

#### Scalene Triangle

A scalene triangle has all the sides of **unequal** length and its three interior **angles** of unequal measure; hence it is an **irregular** polygon.

#### Right-angle Triangle

A right-angle triangle has a **90° **angle which makes one angle **unequal** to the other two **angles**. Also, the **hypotenuse**(the side opposite to the right angle) is the **longest** side, making one side unequal to the other two **sides**.

A right-angle triangle can have all **three** angles and sides **unequal** in measure; therefore is considered an **irregular** polygon.

**Figure 6** shows all the irregular **triangles**.

### Irregular Quadrilaterals

The **irregular** quadrilaterals include rectangles, **trapezoids**, rhombus, parallelograms, and **kites**.

#### Rectangle and Parallelogram

A rectangle and parallelogram have two **opposite** sides of equal length, but not all **sides** are equal, making them **irregular** polygons.

#### Rhombus

A rhombus has all **equal** sides but has only the opposite **angles** equal, not all the angles making it an **irregular** polygon.

#### Trapezoid

A trapezoid is an irregular polygon with two **parallel** sides having all the sides of **unequal** length.

#### Kite

A kite has two **adjacent** sides in pairs of **equal** length which makes it irregular.** Figure 7 **shows all the irregular quadrilaterals.

## A Regular Polyhedron

A regular polyhedron has all its faces as **regular** polygons **congruent** to each other, and an **equal** number of **faces** joined alike at each vertex. For example, a **cube** is a regular polyhedron having congruent **squares** as its faces.

## Example

**Figure 8** shows two hexagons,** A** and **B**.

Which is a **regular** hexagon? What is the order of rotational **symmetry**, and at what **angle** is it rotated to gain **rotational** symmetry? How many **lines** of symmetry are there in a regular hexagon? What is the **sum** of all the **interior** angles of a hexagon?

### Solution

Hexagon **A** is a **regular** hexagon as it has all the **sides** of equal length compared to hexagon **B**.

A regular hexagon’s rotational **order** of symmetry is **six**, which is equal to the number of its **sides**. The **angle** at which the hexagon is rotated to gain **symmetry** is 360°/6 = **60°**.

A regular hexagon has **six** lines of symmetry passing through its **center**.

As an **interior** angle of a regular hexagon equals **120°**, the **sum** of the interior angles is calculated by adding **120°** six times or multiplying 120° by **six**, which gives:

Sum of Interior Angles of a Hexagon = 120° ✕ 6 = **720°**

*All the images are created using GeoGebra.*