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In the intricate dance of **geometric** forms, the **Octagon** has assumed a position of quiet prominence. A figure with eight sides, it is neither as simple as a triangle nor as conventional as a square. Yet, it seamlessly blends simplicity with **complexity**, creating an **equilibrium** that has fascinated mathematicians, architects, and artists alike. Throughout history, the Octagon has symbolized **harmony** and **balance**, manifested in buildings from the 8th-century Umayyad architectural marvels to modern mixed martial arts cages.

This article embarks on a journey into the captivating world of **octagons**, their mathematical principles, their historical significance, and their role in our everyday lives. With each sentence, we will unravel another side of this enigmatic figure, delving deep into the multifaceted aspects of this **eight-sided wonder**. Buckle in and prepare to view your world from an **eight-angled perspective**.

## Definition

A **polygon** having eight sides and eight angles is referred to as an octagon in geometry. The term originates from the **Greek** word “**oktágōnon**,” with “**oktá**” meaning **eight** and **“gōnia”** meaning **angle**.

There are various types of octagons, but the most common one is the regular Octagon, where all the sides and angles are equal. In a regular octagon, each interior angle measures **135 degrees**, and the sum of all interior angles is **1080 degrees**.

**Octagons** are widely used in **architecture**, **design**, and **symbols** due to their aesthetic appeal and geometric properties. An everyday example is the **stop sign**, which is shaped like a regular octagon. Below is the generic diagram of an octagon having eight sides.

Figure-1: Generic octagon.

## Historical Significance

The **Octagon** carries a significant historical and cultural weight in various societies around the world, dating as far back as ancient times. Let’s delve into some key instances that demonstrate the historical significance of the Octagon**.**

**Ancient Architecture**

Many** ancient civilizations** incorporated the Octagon into their architectural designs due to its unique geometric properties, blending the circular with the rectilinear. The **octagonal Tower** of the **Winds in Athens** was built in the second century BC. This structure considered the world’s first meteorological station, was adorned with **sundials**, a water clock, and a **wind vane**.

**Baptistries in Christian Architecture**

In **Christian symbolism**, the number eight often signifies **rebirth** and **Resurrection**, deriving from the tradition of celebrating the Resurrection on the eighth day (Sunday). Consequently, many early **Christian** and **Byzantine baptisteries** were built with an octagonal shape. Notably, the **Lateran Baptistery** in Rome, one of the oldest in the **Christian world**, follows this tradition.

**Islamic Architecture**

The **Octagon** plays a crucial role in Islamic architecture due to its geometric and symbolic significance. The Octagon often serves as a transitional shape between a square base and a circular dome in many **mosques** and **Islamic buildings**. A prominent example is the Dome of the **Rock** in **Jerusalem**, an octagonal building dating back to the 7th century AD, demonstrating the architectural transition from a square base to a **circular dome**.

**Chinese Culture**

In **Chinese** culture, the Octagon has significant importance in **Feng Shui** and **I Ching**. The** Ba Gua**, an octagonal diagram, is used in Feng Shui as a tool to understand how different aspects of life can be affected by a building or a room’s layout. Each of the eight sides corresponds to a different life aspect, such as **wealth**, **family**, or **career**.

**Victorian Architecture**

Due to the advocacy of **phrenologist Orson Squire Fowler**, who felt octagonal homes gave more sunlight and were simpler to heat and cool, octagons briefly gained favour in the United States in the middle of the **19th century**. Though the octagon fad didn’t last long, it did result in some unique examples of Octagon houses that can still be seen today.

**Octagon’s Geometry**

An **octagon**, in its most general sense, is a polygon with eight edges (or sides) and eight **vertices** **(or corners)**. While this term can refer to any** eight-sided polygon** without any qualifications, an** “octagon”** typically refers to a regular octagon – one with sides of equal length and internal angles of equal measure.

In a **regular octagon**, each side is of equal length, and every interior angle is equal to **135 degrees**. To visualize this, consider a square and draw another square of the same size rotated by **45 degrees** around the center point. The overlapping shape you create is a regular octagon. This shape is **equilateral** (all sides are equal in length) and equiangular (all internal angles are equal).

Any **octagon’s inner angles** add up to **1080 degrees**, always, which is calculated by the formula **(n-2) x 180**, where **n** is the number of sides, which is **8** for an octagon. This formula holds true for any polygon, not just octagons.

A regular octagon can be split into eight congruent isosceles triangles. These triangles have a vertex angle of **135 degrees** and two base angles, each of **22.5 degrees**. Also, the regular Octagon can be divided into one square and eight right triangles, offering various possibilities for **geometric problem-solving**.

The **Octagon** also plays a significant role in the field of **tessellation**, which is defined as a group of closely spaced-out forms, particularly polygons in a repetitive pattern without gaps or overlapping. The regular Octagon is not tessellated on its own, but it can form a tessellation when combined with squares or other shapes.

**Octagon Types**

An **octagon**, by definition, is a polygon that has eight sides. However, octagons can vary widely in their shapes and properties based on the lengths of their sides and the **magnitudes** of their angles. Here are some common types of octagons.

**Regular Octagon**

This is the most **symmetrica**l type of **Octagon**, where all sides and all angles are equal. Each interior angle in a regular octagon measures **135 degrees**, and every side is the same length. Below is the generic diagram of a regular octagon having eight sides.

Figure-2: Regular octagon.

**Irregular Octagon**

This is an **octagon** where not all sides and angles are equal. An **irregular octagon** can still be concave or **convex**, but it doesn’t have the same level of symmetry as a regular octagon. Below is the generic diagram of an irregular octagon having eight sides.

Figure-3: Irregular octagon.

**Concave Octagon**

A **concave octagon** has at least one angle that is more than **180 degrees**, giving the shape a **“caved-in”** appearance. This is in contrast to a convex octagon, where all angles are less than **180 degrees**. Below is the generic diagram of a concave octagon having eight sides.

Figure-4: Concave octagon.

### **Convex Octagon**

A **convex octagon** has all its interior angles less than **180 degrees**. Accordingly, no line segment connecting any two points along the boundary ever leaves the polygon. Both regular and irregular octagons can be convex. Below is the generic diagram of a convex octagon having eight sides.

Figure-5: Convex octagon.

**Related Parameters**

In **geometric mathematics**, there are several formulas associated with octagons, especially regular octagons. The following is a list of some of the more noteworthy ones.

**A Regular Octagon’s Perimeter**

The following formula can be used to get a regular **octagon’s perimeter (P)** with equal-length sides.

P = 8 **× **s

where s is the length of one side.

**Area of a Regular Octagon**

The **area** **(A)** of a regular octagon can be calculated using the formula;

$$A = 2 \times s^2 (1+\sqrt{2})$$

where **s **is the length of one side.

**Internal Angles of an Octagon**

The formula for the sum of any polygon’s internal angles is;

Sum = (n-2) **×** 180°

Where **n** is the number of sides, for an octagon, **n** is **8**, so the sum of the interior angles is **1080°**. For a regular octagon, each individual angle is **135°**.

**Exterior Angles of a Regular Octagon**

Any **polygon’s outer angles** add up to** 360°**. For a regular octagon, each individual exterior angle would be **360°/8 = 45°**.

**Diagonals of an Octagon**

The number of **diagonals (D)** in an octagon can be calculated using the formula;

D = n **× **(n-3)/2

where **n** is the number of sides. For an octagon, **n** is **8**, so the number of diagonals is **20**.

**Apothem of a Regular Octagon**

The **apothem** **(a)** of a regular octagon (which is the distance from the center to any side, perpendicular to that side) can be calculated from the side length **(s)** using the formula;

a = s/(2 **× **tan(π/8))

**Octagon Applications**

Certainly, **octagons** are found in a variety of real-world applications, many of which may not even be apparent at first glance. The Octagon is particularly notable in **traffic regulations**, **sports**, **architecture**, and **design** due to its easily recognizable shape and its properties of balance and symmetry. Here’s a more detailed look.

**Traffic Regulations**

**Perhaps the most common use of an octagon** in everyday life is the **stop sign**. In many countries, traffic stop signs are made in the shape of a **regular octagon**. The purpose of this eight-sided figure in traffic regulation is primarily psychological and practical. The unique shape is easily recognizable and stands out among other road signs, allowing drivers to identify it quickly, even at a distance or in low visibility conditions. This prompt recognition is crucial in ensuring the safety and **smooth flow of traffic**.

**Sports**

In the sport of **Mixed Martial Arts** (MMA), contests are often held in an eight-sided cage known as **“The Octagon.”** This design choice isn’t merely aesthetic; it serves a practical purpose. The Octagon provides more space than a traditional **boxing ring**, allowing **athletes** to **maneuver** more freely. Its shape also means there are no corners where a competitor could be trapped. Moreover, the octagonal design provides better sightlines for spectators and officials, ensuring fair and enjoyable viewing.

**Architecture**

The **Octagon** is frequently found in **architecture**, both in the structural design of buildings and in decorative elements. The shape is often used because it combines the strength and stability of squares or rectangles with the aesthetic appeal of circles. For example, the octagonal floor plan has been used in high-profile buildings like the **Palazzo Reale in Naples** or the **Leaning Tower of Pisa**. Additionally, the Octagon is commonly used in the design of less grand structures like gazebos, where it provides a pleasing, symmetrical shape and broad views of the surrounding area.

**Interior Design**

**Octagonal** shapes are often used in **tiles**, **carpets**, and other** interior design elements**. Using octagonal tiles can create interesting and intricate patterns that square or rectangular tiles can’t. Octagonal **mirrors** or wall decorations can also provide a distinctive alternative to the usual **rectangular** or **circular** options.

**Urban Planning**

In city planning, **octagonal buildings** and structures can create efficient, aesthetically pleasing layouts. For example, the **Octagon** on Roosevelt Island in New York was originally designed as a **hospital**. Its unique shape allowed for maximum light and air circulation. Today, it has been converted into a **residential building**, but its octagonal design is still a defining feature.

**Symbolism and Spiritual Practices**

In various cultures and spiritual practices, the **Octagon** holds significant symbolic meaning. In Christianity, the Octagon is often associated with **regeneration**, **rebirth**, and **transition** from earthly to heavenly. Many baptismal fonts in early Christian churches were octagonal for this reason. In Chinese culture, the** Ba Gua**, an octagonal diagram, is used in **Feng Shui** to evaluate energy flow and balance in a space.

**Coding and Computers**

In **computer graphics**, **octagonal** shapes are sometimes used as approximations of circles, as the eight-sided shape can be simpler to generate and manipulate programmatically. In the context of **digital images**, the Octagon can be used as the basis for certain algorithms for** blurring** or for **detecting** edges and shapes within an image.

**Table Games**

**Octagons** are also found in various board games and table games. For example, in the** Settlers of Catan** game, hexagonal tiles are surrounded by octagonal **“ports.”** Octagonal gaming tables are also popular for poker and other card games, providing **equitable** seating positions for numerous players.

**Exercise**

**Example 1**

If the side length of a regular octagon is **5 cm**, what is the **perimeter**?

**Solution**

The perimeter **(P)** of a regular octagon is given by the formula **P = 8**×**s**, where **s** is the length of a side.

P = 8 **×** 5 = 40 cm.

So, the perimeter of the Octagon is **40 cm**.

**Example 2**

What is the **area** of the octagon given in Figure-6?

Figure-6.

**Solution**

The formula for the area **(A)** of a regular octagon is $A = 2 \times s^2 (1+\sqrt{2})$, where **s** is the length of a side.

$$A = 2 \times 3^2 (1+\sqrt{2})$$

$$A = 2 \times 9 (1+\sqrt{2})$$

$$A = 18(1+\sqrt{2})$$

$$A = 18+18\sqrt{2}$$

So, the area of the Octagon is $A = 18(1+\sqrt{2})$ **cm²**.

**Example 3**

How many **interior** **angles** are there in a standard octagon, and what are their measurements?

**Solution**

An octagon’s internal angles add up to **1080 degrees**. Since a regular octagon has equal angles, each angle measures **1080° / 8 = 135°**.

So, each interior angle of a regular octagon is **135°**.

**Example 4**

How many **outside (exterior) angles** are there in a normal octagon, and what are their measurements?

**Solution**

Solution: The sum of the exterior angles for any polygon is **360°**. For a regular octagon, each individual exterior angle would be **360° / 8 = 45°**.

So, each exterior angle of a regular octagon is **45°**.

**Example 5**

How many **diagonals** does an octagon have?

**Solution**

The number of diagonals **(D)** in a polygon is given by the formula **D = n(n-3)/2**, where **n** is the number of sides. For an octagon, **n** is **8**.

D = 8 **× **(8 – 3)/2

D = 8 **×** 5/2

D = 20

So, an octagon has **20 diagonals**.

**Example 6**

Apothem of a Regular Octagon: If the side length of a regular octagon is **4 cm**, what is the **length** **of the** **apothem**?

**Solution**

The formula for the apothem **(a)** of a regular octagon is;

a = s/(2 **× **tan(π/8)),

where** s** is the side length. By putting the value of **s**, we get

a = 4/(2 **× **tan(π/8))

a = 4/(2 **× **0.414)

a = 4.83 cm (approximately)

So, the apothem of the Octagon is approximately **4.83 cm**.

**Example 7**

If the **area** of a regular octagon is **48 cm²**, what is the **length** **of one side**?

**Solution**

Using the formula for the area of an octagon $A = 2 \times s^2 (1+\sqrt{2})$, we solve for **s**.

$$48 = 2 \times s^2 (1+\sqrt{2})$$

Dividing both sides by 2(1 + √2) gives us

$$s^2 = \frac{48}{2 \times (1+\sqrt{2})}$$

Taking the square root of both sides gives us

$$s = \sqrt{\frac{48}{2 \times (1+\sqrt{2})}}$$

s = 2.41 cm (approximately)

So, the side length of the Octagon is approximately **2.41 cm**.

**Example 8**

If the side **length** of a regular octagon is** 2 cm**, What is the **circumradius** (the circumscribed circle’s radius)?

**Solution**

The formula for the circumradius **R** is **R = s/(2×sin(π/n))**, where **n** is the number of sides and **s** is the side length.

R = 2 / (2 **× **sin(π/8))

R = 2 / (2 **× **0.383)

R = 2.61 cm (approximately)

So, the circumradius of the Octagon is approximately **2.61 cm**.

*All images were created with GeoGebra.*