Area of an Octagon: Meaning, Formula, Steps & Solved Questions

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Octagon can be defined as an eight-sided polygon with eight angles. Octagons can be classified into two different types based on their side lengths- Regular Octagon and Irregular Octagon. The octagons that have an equal length of sides and all the same value angles are called regular octagons and the octagons with unequal sides and angles are known as irregular octagons. As an octagon has eight sides, the sum of its internal angles is 1080°. The measure of each interior angle in a regular octagon is 135° and the exterior angles of a regular octagon measure 45° each. The area of an octagon is the total space/area that has been enclosed by all eight sides of the octagon. The general formula for calculating the area of an octagon is 2s²(1+√2), where s denotes the length of the side of the octagon.

Table of Contents

Types of Octagon
Area of An Octagon
Area of Octagon Formula
How to Calculate The Area of An Octagon?
Things to Remember
Sample Questions

Key Terms: Octagon, Regular Octagon, Irregular Octagon, Interior Angles, Exterior Angles, Area, Perimeter, Vertex, Diagonals, Parallelograms, Rectangles, Triangles

Types of Octagon

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There are two types of octagons. They can be regular or irregular.

Regular Octagon: A regular octagon has sides with the same length and all the same value angles. A regular octagon has all sides and angles equal. Each interior point in a regular octagon measures 135°, summing up to 1080° degrees. Each exterior angle measures 45° in an octagon, giving a total sum of 360°. There is a total of 20 diagonals that are present in a regular octagon.
Irregular Octagon: An irregular octagon is an octagon that is not a regular octagon. Irregular octagon has unequal sides and/or angles.

what is octagon

Octagon

Area of An Octagon

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An octagon is a closed shape that has 8 sides and thus 8 angles. The interior angles of a regular octagon generally measure 135° each, while the exterior angles are 45° each. There are various kinds of octagons yet every one of them have 8 sides and 8 angles. The most widely recognized type is a regular octagon since it has equivalent side lengths and equal angles.

The area of an octagon is a proportion of the area canvassed by an octagon in a two-layered plane. The region is the space encased by any mathematical shape. In basic words, we can say that the area of an octagon is the area covered by a two-layered figure of an octagon in a plane.

what is regular and irregular octagon

Regular and Irregular Octagon

Area of Octagon Formula

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The formula for calculating the area of an octagon is given as follows:

Area of an Octagon = 2s²(1+√2)

Here, s is the length of the side of the octagon

The area of an octagon is characterized as the complete space enclosed within the limit of an octagon. The unit for the area of an octagon is square units.

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Types of Quadrilaterals	Quadrilaterals	Types of Triangles
Difference Between Rhombus and Parallelogram	Quadrilateral Formulas	Area of a Trapezoid Formula with Examples

How to Calculate The Area of An Octagon?

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The area of an octagon can be determined by utilizing the area formula for octagon using the below-mentioned steps:

As we know, Area of an Octagon = 2s²(1+√2)

Step 1: Find out and calculate the length of the side of the octagon.
Step 2: Now, calculate the square of its length.
Step 3: Calculate the product of the square of its length to 2(1+√2). With this, we can get the area of the octagon.
Step 4: Substituting the values in the area of an octagon formula 2s²(1+√2), in order to get the answer.
Step 5: Depict the answer received in square units.

Read More: Area of Square Using Diagonal

Things to Remember

Octagon is a closed shape that has eight sides and thus eight angles.
An octagon can be a regular octagon or an irregular one.
A regular octagon is a two-dimensional closed figure with eight equal sides, eight equal interior angles and eight vertices. The internal angle of any regular octagon is 1350 each.
An irregular octagon is one that does not have equal sides or equal angles.
The area of an octagon is the sum of the areas of the eight triangles that make up the shape. There are a number of different ways to find the area of a regular octagon, such as breaking it down into parallelograms and rectangles or breaking it down into trapezoids and triangles.
The general formula to calculate the area of an octagon is 2s²(1+√2), where s denotes the length of the side of the octagon.

Sample Questions

Ques. Find the area of an octagon of a side 10 cm. (3 Marks)

Ans. Given,
Side of an octagon (a) = 10 cm

Area of an octagon
= 2×a²× (1 + √2)
= 2× 10²× (1 + √2) cm²
= 2 × 100× (1 + 1.414) cm²
= 482.8 cm²

Ques. The perimeter of an octagonal stop signboard is 48 cm. Find the area of the signboard. (3 Marks)

Ans. Given,

Now, the perimeter of the stop signboard is given as 48 cm

Perimeter of an Octagon = 8a

48 cm = 8a

a = 48/8 = 6 cm

Area of an Octagon = 2s²(1+√2)

Thus, area of the stop signboard = 2* 6*6 *2.414

= 173.81 cm sq.

Ques. Find the area of the octagon if the length of the side of the octagon is 14 inches. (3 Marks)

Ans. Length of the side, s = 14 in

Through the formula for the area of the octagon, we get

A = 2s²(1+√2)

A = 2 ×14²(1+√2)

A = 946.37

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

Ques. Anjali was told that the area of an octagon was 25.54 units square. Help him to calculate the length of the side of the octagon? (3 Marks)

Ans. Area of the octagon, A = 25.54 square units

We will be calculating the area of the octagon using the formula stated above,

A = 2s²(1+√2)

25.54 = 2 × s²(1+√2)

s = 2.3 units,

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

Ques. Find the area of a regular octagon if its side measures 5 units. (3 Marks)

Ans. As we know that the sides of a regular octagon are of equal length, thus,

The length of the side (a) = 5 units.

Now, we can calculate the area of the octagon using the formula,

Area of a Regular Octagon = 2a²(1 + √2).

Substituting the value of 'a' in the formula, we get,

We know that, Area of a Regular Octagon is given as 2a²(1 + √2) = 2 × (5)²× (1 + √2)

= 50 × (1 + √2) = 120.71 square units.

Hence, the area of the octagon is calculated as 120.71 square units

Ques. Find the area of a regular polygon with a side of 2.5cm. (3 Marks)

Ans. Given that,

The side of the octagon is 2.5

Area of an octagon = 2a²(1 + √2)

A = 2(2.5)²(1 + √2)

Ques. Calculate the area and perimeter of a regular octagon whose side is 2.3 cm. (3 Marks)

Ans.Given, side of the octagon = 2.3 cm

Area of an Octagon = 2a²(1 + √2)

Area of an Octagon = 2 × 2.3 ²(1 + √2)

= 25.54 cm²

Therefore, the perimeter of the octagonwill be 8a=8 × 2.3 = 18.4 cm

Ques. The perimeter of an octagonal stop signboard is 32 cm. Find the area of the signboard. (3 Marks)

Ans. Given,

Now, the perimeter of the stop signboard is 32 cm

Perimeter of an Octagon = 8a

32 cm = 8a

a = 32/8 = 4 cm

Area of an Octagon = 2a²(1 + √2)

Area of the stop sign board = 2 × 2 ²(1 + √2)

= 77.248 cm²

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Area of an Octagon: Meaning, Formula, Steps & Solved Questions

Types of Octagon

Area of An Octagon

Area of Octagon Formula

How to Calculate The Area of An Octagon?

Things to Remember

Sample Questions

CBSE X Related Questions

1.
Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

2.
Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent.
Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

3.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

4.
Use graphical method to solve the system of linear equations : $x = -3$ and $5x - 2y = -5$.

5.
In the given figure, point D divides the side BC of $\Delta ABC$ in the ratio $1 : 2$. Find length AD.

6.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

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Area of an Octagon: Meaning, Formula, Steps & Solved Questions

Types of Octagon

Area of An Octagon

Area of Octagon Formula

How to Calculate The Area of An Octagon?

Things to Remember

Sample Questions

CBSE X Related Questions

1.Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

2.Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent. Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

3.An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

4.Use graphical method to solve the system of linear equations : $x = -3$ and $5x - 2y = -5$.

5.In the given figure, point D divides the side BC of $\Delta ABC$ in the ratio $1 : 2$. Find length AD.

6.PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

2.
Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent.
Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

3.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

4.
Use graphical method to solve the system of linear equations : $x = -3$ and $5x - 2y = -5$.

5.
In the given figure, point D divides the side BC of $\Delta ABC$ in the ratio $1 : 2$. Find length AD.

6.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.