**Octagon**

The total area that the octagon’s eight sides enclose is referred to as the area of the octagon. An eight-sided polygon with eight interior and exterior angles makes up this shape. By splitting the octagon into 8 equal isosceles triangles, the area of the shape can be determined.

**What is the Area of Octagon?**

The area of an octagon, a 2-dimensional object with 8 sides, is the space inside the shape’s 8 sides. We can utilise the area of an isosceles triangle to get the octagon’s area. We determine the area of the shape by dividing it into eight equal isosceles triangles. The length of the sides and the angles between them are both equal in a regular octagon. Each inner angle is 135 degrees in length, whereas each external angle is 45 degrees.

**Area of Octagon Formula**

The following formula is used to determine an octagon’s area:

2s2(1+2), where s is the octagon’s side length.

**How to calculate Area of Octagon?**

Octagon’s surface area is 2s2(1+2). We may locate the octagon’s area by using the steps listed below.

First, determine how long each side of the octagon is.

Step 2: Locate the square with the longest side.

Step 3: Calculate the product of the length’s square with 2(1+2). This will reveal the octagon’s area.

Step 4: To find the answer, substitute the appropriate values into the area of an octagon formula 2s2(1+2).

Step 5: Display the solution in square units.

**Octagon Formulas**

Area of an Octagon 2a2(1+√2)

Perimeter of an Octagon 8a |

**Derivation of Octagon Formulas:**

Think of a standard octagon with “a” units on each side.

The region occupied inside the octagon’s boundary is what is meant by the formula for an octagon’s area.

We break an octagon into eight little isosceles triangles to compute its area. To determine the overall area of the polygon, multiply the size of one of the triangles by eight.

**Formula for Perimeter of an Octagon:**

The length of an octagon’s boundary is referred to as its perimeter. Therefore, the perimeter will equal the total length of all sides. The formula for an octagon’s perimeter is as follows:

8-sided length times the perimeter

So, an octagon’s circumference is equal to 8a

**Properties of Regular Octagon:**

- There are eight sides and eight angles in a regular octagon.

- Each side’s length and each angle’s measurement are the same.

- A regular octagon has a total of 20 diagonals.

- Each inside angle is 135 degrees in length, so the total interior angle is 1080 degrees.

- Each external angle measures 45 degrees, hence the total number of exterior angles is 360.

**Question 1**

**we need to find the area and perimeter of a regular octagon with a side length of 2.5 cm.**

The formula for the perimeter of a regular octagon is P = 8s, where s is the length of one of the sides. Substituting s = 2.5 cm into this formula, we get:

P = 8 × 2.5 = 20 cm

So the perimeter of the octagon is 20 cm.

To find the area of a regular octagon, we can use the formula A = 2(1 + √2)s², where s is the length of one of the sides. Substituting s = 2.5 cm into this formula, we get:

A = 2(1 + √2)(2.5)² ≈ 43.30 cm²

Therefore, the area of the regular octagon is approximately 43.30 cm².

**Question 2**

**The problem requires us to find the area of a stop signboard which is in the shape of a regular octagon, given that its perimeter is 37 cm.**

To begin solving the problem, we can use the formula for the perimeter of a regular octagon which is P = 8s, where s is the length of one of its sides. Substituting P = 37 cm, we get:

37 = 8s

Solving for s, we get:

s = 4.625 cm

So the length of one side of the octagon is 4.625 cm.

Next, we can use the formula for the area of a regular octagon, which is A = 2(1 + √2)s², to find the area of the signboard. Substituting s = 4.625 cm, we get:

A = 2(1 + √2)(4.625)² ≈ 124.84 cm²

Therefore, the area of the stop signboard is approximately 124.84 cm².

**What are the formulas for calculating the octagon’s area?**

Octagon area is equal to 2s2(1+2) = 4sa.where s is an octagonal side and an is an apotheosis.

**How many diagonals can be created within an octagon?**

In a normal octagon, there are 20 total diagonals.

**What does the octagon represent?**

The octagon is occasionally used as a representation of infinity. The octagon has been compared to a circle trying to become a square and a square trying to become a circle.