In the realm of polygons, the octagon stands tall with its eight sides and angles. This geometric figure, derived from the Greek words “octa” (meaning eight) and “gonia” (meaning angle), possesses a symmetry and balance that make it a captivating element in the world of shapes. Let’s embark on a brief exploration of the octagon, unraveling its defining features, properties, and real-world applications.

**What is an Octagon?**

An octagon is a polygon with eight sides and eight angles. Its structure is characterized by straight line segments connecting eight vertices, creating an enclosed figure.

**Types of Octagons**

Octagons, being eight-sided polygons, can vary in their characteristics and properties. Here are the main types of octagons based on their features:

### 1. **Regular Octagon**

- All sides and angles are equal.
- Each interior angle measures 135 degrees.
- Exhibits rotational symmetry.

### 2. **Irregular Octagon**

- Sides and/or angles are not equal.
- Varied side lengths and angle measures.
- Lacks the symmetry found in a regular octagon.

### 3. **Convex Octagon**

- No interior angle is greater than 180 degrees.
- All angles point outward.
- Diagonals remain inside the shape.

### 4. **Concave Octagon**

- At least one interior angle is greater than 180 degrees.
- Has at least one “caved-in” corner.
- Diagonals extend outside the shape.

### 5. **Regular Irregular Octagon**

- A non-convex octagon with equal sides and angles.
- Exhibits characteristics of both regular and irregular octagons.
- Maintains symmetry but has some unequal features.

**Properties of Octagons**

Octagons, being eight-sided polygons, possess several properties that define their geometric characteristics. Here are the key properties of octagons:

- An octagon has 8 sides and 8 angles.
- The sum of interior angles is 1080 degrees.
- In a regular octagon, each angle is 135 degrees.
- In a regular octagon, all sides are of equal length.
- The sum of exterior angles is always 360 degrees.
- An octagon has 20 diagonals.
- In a regular octagon, the length of each diagonal can be calculated using a formula.
- A regular octagon has rotational symmetry of 45 degrees.

**Perimeter of an Octagon**

The perimeter (P) of an octagon, which is the total length of its eight sides, can be calculated using a simple formula:

P = \(\text{Sum of all side lengths}\)

If the octagon is regular (all sides are equal), the formula becomes:

P = 8 \(\times \text{Length of one side}\)

In the case of an irregular octagon with different side lengths, you would sum up the lengths of all eight sides.

For example, if you have a regular octagon with each side measuring 5 units, the perimeter (P) would be:

P = 8 \(\times 5 = 40 \text{ units}\)

If you have an irregular octagon with side lengths 6, 7, 6, 7, 6, 7, 6, and 7 units, the perimeter would be:

P = \(6 + 7 + 6 + 7 + 6 + 7 + 6 + 7 = 52 \text{ units}\)

In summary, the perimeter of an octagon is found by adding up the lengths of its individual sides, and for a regular octagon, it’s simply 8 times the length of one side.

**Solved Examples on Octagon**

**Example 1:Consider a regular octagon with each side measuring 6 units.**

**Solution:**

Perimeter (P) = \( 8 \times \text{Length of one side}\)

P = \( 8 \times 6 = 48 \text{ units}\)

The formula for the area of a regular octagon is given by:

Area (A) = \( \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}\)

For simplicity, assuming the apothem is also 6 units (since it’s a regular octagon):

A = \( \frac{1}{2} \times 48 \times 6 = 144 \text{ square units}\)

**Example 2: Consider an irregular octagon with side lengths 8, 7, 8, 7, 8, 7, 8, and 7 units.**

**Solution:**

Perimeter (P) = \(\text{Sum of all side lengths}\)

P = \(8 + 7 + 8 + 7 + 8 + 7 + 8 + 7 = 60 \text{ units}\)