A polygon is a two-dimensional (2-D) closed figure made up of straight line segments. In geometry, the octagon is a polygon with 8 sides. If the lengths of all the sides and the measurement of all the angles are equal, the octagon is called a regular octagon. In other words, the sides of a regular octagon are congruent. Each of the interior angle and the exterior angle measure 135° and 45° respectively, in a regular octagon. There is a predefined set of formulas for the calculation of perimeter, and area of a regular octagon which is collectively called as octagon formula. For an octagon with the length of its edge as “a”, the formulas are listed below.

Octagon formula helps us to compute the area and perimeter of octagonal objects.

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## Derivation of Octagon Formulas:

Consider a regular octagon with each side “*a” *units.

**Formula for Area of an Octagon:**

Area of an octagon is defined as the region occupied inside the boundary of an octagon.

In order to calculate the area of an octagon, we divide it into small eight isosceles triangles. Calculate the area of one of the triangles and then we can multiply by 8 to find the total area of the polygon.

Take one of the triangles and draw a line from the apex to the midpoint of the base to form a right angle. The base of the triangle is ** a**, the side length of the polygon and OD is the height of the triangle.

Area of the octagon is given as 8 x Area of Triangle.

2 sin²θ = 1- cos 2θ

2 cos²θ = 1+ cos 2θ

**Formula for Perimeter of an Octagon:**

Perimeter of an octagon is defined as the length of the boundary of the octagon. So perimeter will be the sum of the length of all sides. The formula for perimeter of an octagon is given by:

Perimeter = length of 8 sides

**So, the perimeter of an Octagon = 8a**

**Properties of a Regular Octagon:**

- It has eight sides and eight angles.
- Lengths of all the sides and the measurement of all the angles are equal.
- The total number of diagonals in a regular octagon is 20.
- The sum of all interior angles is equal to 1080 degrees, where each interior angle measures 135 degrees.
- The sum of all exterior angles is equal to 360 degrees, where each exterior angle measures 45 degrees.

**Solved examples Using Octagon Formula:**

**Question 1:** Calculate the area and perimeter of a regular octagon whose side is 2.3 cm.

**Solution:** Given, side of the octagon = 2.3 cm

Perimeter of the octagon **= **8a **= **8 × 2.3 = 18.4 cm

**Question 2: **Perimeter of an octagonal stop signboard is 32 cm. Find the area of the signboard.

**Solution**: Given,

Perimeter of the stop sign board = 32 cm

Perimeter of an Octagon = 8a

32 cm = 8a

a = 32/8 = 4 cm

## Area Of An Octagon Formula

You know that polygons are 2D closed figures which are made up of 3 or more line segments. Triangle is the smallest polygon with 3 sides. Similarly, the polygon which has 8 sides is called an octagon. If the 8 sides of the octagon are equal in length and the angles between these sides are equal, the octagon is called a regular octagon. In a regular octagon, the measure of each of the interior angle is 135 degrees. The exterior angles measure 45 degrees each.

In this section, you will learn how to calculate the area of an octagon. Also, how to derive a formula for the same and solve some examples so as to give you a clear understanding of the concept.

Now, the formula for computing the area of a regular octagon can be given as:

Where ‘a’ is the length of any one side of the octagon.

Click here to learn more about octagon and its types in detail.

**Properties of a Regular Octagon:**

Before we go ahead and derive the area of an octagon formula, let us go through some of the basic properties of a regular octagon.

- It has 8 sides and 8 interior angles.
- All the sides are equal in length and all the angles are equal in measurement.
- There are a total of 20 diagonals in a regular octagon.
- Each interior angle measures 135 degrees. Hence, the sum of all interior angles is (135 x 8) = 1080 degrees.
- Each exterior angle measures 45 degrees. Hence, the sum of all exterior angles is (45 x 8) = 360 degrees.

**Derivation:**

For deriving the area formula, let us consider a regular octagon. Each of its sides is of length *a *units. Mathematically, area of an octagon is defined as the region covered inside the boundary of an octagon.

Let us divide the octagon into 8 small isosceles triangles by joining each vertex to the centre. Now if we compute the area of one of these triangles and multiply it by 8, we will obtain the area of the octagon.

Take any one of the triangles and draw a line from the centre O of the hexagon to the midpoint of the base AB. Name the midpoint as D. You will get a right-angled angle. The base length of the triangle is ** a** since it is nothing but the side of the octagon. Thus, OD forms the height of the triangle OAB.

Area of the octagon is calculated by summing up the areas of the 8 triangles, or, 8 x Area of one triangle (OAB).

We know the relation 2 sin²θ = 1- cos 2θ. We can write it as:

**Solved Examples**

**Question 1:** Calculate the area of a regular octagon whose side is 35 cm.

**Solution:** Given,

Side of the octagon = 35 cm

## What is the Area of Octagon?

An octagon is a 2-dimensional shape with 8 sides and the area is the space within the 8 sides of the shape. To calculate the area of the octagon we can use the area of an isosceles triangle. We divide the area of the shape into eight equal isosceles triangles and calculate it accordingly. A regular octagon has the length of the sides equal and the angles between these sides are equal. The measure of each interior angle is 135° and the exterior angles measure 45° each.

### Area of Octagon Formula

**The general formula we use to calculate the area of the octagon is:**

**2s ^{2}(1+√2), **where s is the length of the side of the octagon.

### How to Calculate Area of Octagon?

The area of an octagon is 2s^{2}(1+√2). By using the following steps mentioned below we can find the area of the octagon.

**Step 1**: Calculate the length of the side of the octagon.**Step 2**: Find the square of the length of the side.**Step 3**: Find out the product of the square of its length to 2(1+√2). This will give the area of the octagon.**Step 4**: By substituting the respective values in the area of an octagon formula 2s^{2}(1+√2) we will get the answer.**Step 5**: Represent the answer in square units.

## Area of Octagon Examples

**Example 1:** Find the area of the octagon if the length of the side of the octagon is 14 in.

**Solution:**

Length of the side, s = 14 in

Using the formula for the area of the octagon,

A = 2s^{2}(1+√2)

A = 2 ×14^{2}(1+√2)

A = 946.37

Therefore, the area of the octagon is 946.37 square inches.

**Example 2:** Walter was given an area of an octagon as 25.54 units square. Can you help him find the length of the side of the octagon?

**Solution:**

Area of the octagon, A = 25.54 units square.

Using the formula for the area of the octagon,

A = 2s^{2}(1+√2)

25.54 = 2 × s^{2}(1+√2)

s = 2.3 units

Therefore, the length of the side of the octagon is 2.3 units.

**Example 3:** Find the area of a regular octagon if its side measures 5 units.

**Solution:**

The sides of a regular octagon are of equal length. Here, the side length, a = 5 units. We can find the area of the octagon using the formula, Area of a Regular Octagon = 2a^{2}(1 + √2). Substituting the value of ‘a’ in the formula, we get, Area of a Regular Octagon = 2a^{2}(1 + √2) = 2 × (5)^{2} × (1 + √2) = 50 × (1 + √2) = 120.71 square units. Therefore, the area of the octagon is 120.71 square units.

## FAQs on Area of Octagon

### What is the General Formula of Area of Octagon?

If the shape is a polygon and it has eight sides, we call it an octagon. To find the area of an octagon we use the following formula. Area of octagon formula = 2 × s^{2} × (1+√2). Where “s” denotes the length of the side of an octagon.

### What is the Area of an Octagon With an Apothem?

Apothem is the line segment drawn perpendicular to the side of the octagon from the center of the octagon. The area of an octagon with an apothem of length l is 8l^{2}(√2-1).

### What is the Area of an Octagon with a Radius?

The area of an octagon with a radius of length r is 2√2r^{2}. Where “r” is the radius of an octagon.

### What is the Area of an Octagon with Side Length “a”?

The area of an octagon with side length a is 2a^{2}(1+√2). Where “a” denotes the length of the side of an octagon.

### What is the Area of an Octagon With Side 5 units?

Area of octagon = 2s^{2}(1+√2)

= 2×5^{2}(1+√2)

= 120.7 units^{2}

Area of the octagon is 120.7 units^{2}.

## What Is Octagon Formula?

The octagon formula is used to calculate the area, perimeter, and diagonals of an octagon. To find the area, perimeter, and diagonals of an octagon we use the following octagon formulas.

### Formulas for Octagon:

To find the area of an octagon we use the following formula: Area of octagon formula = 2 × s^{2} × (1 + √2)

To find the perimeter of an octagon we use the following formula: Perimeter of octagon = 8s

To find the number of diagonals of an octagon we use the following formula: Number of Diagonals = n(n – 3)/2 = 8(8 – 3)/2 = 20

where,

- s = side length
- n = number of sides

## Examples Using Octagon Formula

**Example 1:** Calculate the perimeter and area of an octagon having a side equal to 4 units using the octagon formula.

**Solution:**

To Find: Perimeter and Area

Given: s= 4 units.

Using the octagon formula for perimeter

Perimeter(P) = 8s

P = 8 × 4

P = 32 units

Using the octagon formula for area

Area of octagon = 2s^{2}(1 + √2)

= 2 × 4^{2}(1 + √2)

= 77.25483 units^{2}

**Answer:** Perimeter and area of the octagon are 32 units and 77.25483 units^{2}.

**Example 2:** An octagonal board has a perimeter equal to 24 cm. Find its area using the octagon formula.

**Solution:**

To Find:** ** Area of the octagon.

Given: Perimeter = 24 cm.

The perimeter of octagon = 8s

24 = 8 s

s = 3 cm.

Using the octagon formula for area,

Area of octagon = 2s^{2}(1 + √2)

= 2 × 3^{2}(1 + √2)

= 43.45 cm^{2}

**Answer: **Area of octagonal board is 43.45 cm^{2}.

## FAQs (Frequently Asked Questions)

**Q1. What is the Difference Between a Regular Octagon and an Irregular Octagon?**

Answer: These octagons are differentiated on the basis of the following i.e.:

**Regular Octagon:**A type of octagon that has all sides and angles equal. Each angle of a regular octagon measures 135° each.**Irregular Octagon:**A type of octagon that has all sides and all angles not equivalent. Angle measurement of an irregular octagon is not similar but remembers that they all sum to 1080°.

**Q2. What is the Difference Between a Convex and Concave Octagon?**

Answer: Octagon is that type of polygon which has 8 sides and 8 vertices. An octagon, like any other polygon, can also be further classified either convex or concave. In a convex octagon, none of its interior angles is greater than 180°. On the hand, a concave octagon consists of one or more of its interior angles greater than 180°. Having only the sides equal is not sufficient, in that context, seeing that the octagon may be concave, with equal sides but unequal interior angles. Such octagons are known as equilateral.

**Q3. What is the Bounding Box Related to Octagon?**

Answer: The bounding box of a planar shape is the smallest rectangle which circumscribes the shape entirely. The dimensions of this rectangle are typically described by the height and the width of the shape. The below-given depiction defines the bounding box of a regular octagon. Both the height and the width are described as the distance between two opposite edges. Because of the symmetry, height and width are equal, and as an outcome, the bounding box is actually a square.

## FAQs (Frequently Asked Questions)

**1. What is the area of an octagon?**

The area of an octagon = 2a2 (1 + √2). The details on how to find the area of an octagon are discussed in the study notes provided by Vedantu. Along with calculating the area of an octagon, students can learn about various other concepts involving the other aspects of an octagon like the perimeter and length of an octagon.

**2. What are the properties of an octagon?**

Following are the properties of a regular octagon-

- It consists of 8 sides and 8 angles
- All the sides of an octagon are equal
- All the angles of an octagon are equal
- A regular octagon mainly consists of a total of 20 diagonals.
- The interior angle of the octagon measures 1350, which is a total of 10800
- Each of a regular octagon’s exterior angles measures 45°, giving a total sum of 360°.

**3. What is the perimeter of an octagon?**

The perimeter of an octagon is- Perimeter of octagon = 8a, where a is the given polygon’s each side length. The perimeter mainly consists of the outline or the periphery of an octagon. It can be calculated as the total length of its boundaries. This means that the perimeter of an octagon is the sum of all its sides. Calculating the perimeter of an octagon is explained in brief in the study notes provided by Vedantu. These notes will help students to get a strong mathematical foundation.

**4. Where can I find the study material based on the area of an octagon?**

The study material based on the area of an octagon can be easily found on Vedantu’s website, the link can be accessed easily through the internet by visiting Vedantu. The study notes are available in a pdf format. These notes are based on the CBSE curriculum and are prepared through extensive research and critical analysis of previous year’s question papers.

**5. Are the numerical questions on the area of an octagon difficult to solve?**

The numerical questions based on the area of an octagon are not at all difficult to solve. Before solving the numerical, it is advisable for students to get a good grasp of the various concepts that are discussed in the chapter. This will enable students to get a clearer understanding of the equations and formulas which will then enable them to solve the numerical in an easy and efficient manner.

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