Octagon Formula For Area and Perimeter With Derivation

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In the branch of geometry, the octagon is basically a polygon with eight sides and eight angles. A polygon can be defined as a flat structure that is two-dimensional and is a formation of multiple line segments.

The word octagon is derived from a Greek word that means ‘eight angles’.
It consists of 20 diagonals that meet at the centre of the figure.
The measurements of the exterior and internal angles of the two-dimensional figure are 135° and 45°, respectively.
A 'STOP' sign that is used for a roadblock is a perfect example of an octagon.
The area and perimeter of an octagon are given as:

A= 2a²(1+√2)

P = 8a

where a is the side of the figure

Table of Content

What is an Octagon?
Octagon Formulas
Properties of an Octagon
Derivation of Octagon Formulas
Things to Remember
Sample Questions

Key Terms: Octagon, Octagon Formula, Polygon, Area of Octagon, Perimeter of Octagon, Regular Octagon, Irregular Octagon, Angles, Interior Angles, Exterior Angles

What is an Octagon?

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Octagon is a two-dimensional figure that has eight sides. All the diagonals of the figure are of equal length. Regular octagons and Irregular octagons are two types of octagons.

When all the sides and angles of an octagon are equal, it can be counted as a Regular Octagon.
On the contrary, when the sides and angles of an octagon are different in terms of measurements, it is counted as an irregular octagon.
The sum of interior and exterior angles is 135 degrees and 108 degrees.
Umbrella is the most common example of an octagon.

Octagon

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Area of Rectangle	Areas Related to Circle	Geometry Formula
Tangent to a Circle	Important Questions on Circle	Angle Formula

Octagon Formulas

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Let’s have a look at the chart below that illustrates the important octagon formula. The calculation is performed using the side of the figure.

Category	Formula
Area of an Octagon	2a² (1+√2)
Perimeter of an Octagon	8a
Number of Diagonals	n(n – 3)/2

Example of Octagon Formulas

Example: Calculate the perimeter and area of an octagon having a side equal to 5 units using the octagon formula.

Ans: Given: s= 5 units.
Using the octagon formula for area and perimeter
Perimeter(P) = 8s
P = 8 × 5
P = 40 units
Using the octagon formula for area
Area of octagon = 2s²(1 + √2)
= 2 × 5²(1 + √2)
= 120.71 units²

Properties of an Octagon

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The properties of an octagon are as follows:

One of the most prominent properties of any octagon is that it has eight sides, i.e., eight connected straight lines.
In the case of a regular octagon, the lines and the connected angles are equal.
The measurements of the lines and angles are not equal in the case of an irregular octagon.
For a regular octagon, all the interior angles would be 135°, while the exterior angles would be 45°.
Therefore, the total of the interior angles would be 108°, and the sum of exterior angles would be 360°.
Evidently, as we know, the angles are not equal in the case of the irregular octagon; the total measurement will remain the same as the regular one.

Derivation of Octagon Formulas

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The derivation of octagon formulas are as follows:

Area of an Octagon

The area of any geometrical shape can be defined as the total amount of space that the perimeter of a two-dimensional shape contains. As mentioned earlier, the area of an octagon can be measured by using the formula- 2a ² (1+√2).

If you look carefully, you can see that an octagon can include other geometrical shapes as well.
For example, you can identify four rectangles, four triangles around the corners of the octagon, and a square in the middle of the octagon.
We also know that the area of a square is a².
If we go by the Pythagoras Theorem, all the triangles that are included into an Octagon, have sides- a, a/√2, and a/√2.
As a result each triangles have the area of a²/4
Now, we are left with the rectangles.
The area of the rectangles can be measured as a x √2
All these things considered, the area of an Octagon will be 2a ² (1+√2)
In other words, the area of an octagon = 8 x area of a triangle
8 x a²/4 (1+ √2)
Finally, after we calculate 2a² (1+√2)

Area of Octagon

Perimeter of an Octagon

According to the oxford language dictionary, a perimeter is defined as the continuous line forming the boundary of a closed geometric figure. By far it is clear for us that an octagon has 8 sides.

Hence, the perimeter of an octagon can be easily measured via the formula- 8a (where, a = side length)

Also check:

Related Concepts
Circles	Similarities of Triangle	Geometry Formula
Circumference of a Circle	Perimeter of a Circle	Congruence of Triangles

Things to Remember

A polygon is a 2D geometrical structure with a formation of multiple line segments.
An octagon is a polygon with eight lines and angles
For a regular octagon, all lines and angles are equal.
In the case of an irregular octagon, the lines and angles are not equal.
The area and perimeter of an octagon are given as 2a² (1+√2) and 8a, respectively.

Sample Questions

Ques. Explain the concept of an Octagon with a real life example? (3 marks)

Ans. Octagon is a flat, two-dimensional geometrical structure with eight sides and eight angles. It is basically a polygon with eight sides. The word- octagon came from a Greek phrase which means ‘eight angles’.

If we add up the interior angles of an octagon, the sum will be- 108°, and if we try to add all the exterior angles of an octagon, it will be- 360°.
A real life example of the octagon is the houses and rooms.

Ques. State the differences between regular and irregular octagon? (3 marks)

Ans. The differences between a regular octagon and an irregular octagon can be clearly described through the chart below.

Regular Octagon	Irregular Octagon
All the sides of a regular octagon are equal. All the angles of a regular are equal. A regular octagon looks like the picture below-	The sides of an irregular octagon are not equal. The angles of an irregular octagon are not equal. This is a picture of an irregular octagon-

Ques. State the formula of measuring the perimeter of an octagon with an example? (3 marks)

Ans. The formula for calculating the Perimeter of an Octagon is- 8a

Where a is the side length.
For example, there is a regular octagon, with each side length of 6cm.
Therefore, a = 6cm
Now, as we know, the Perimeter of the Octagon = 8a
So in this particular case the perimeter will be- (8 x 6)cm= 48cm.

Ques.Calculate the area and perimeter of a regular octagon whose side is 4cm? (3 marks)

Ans. As we know the formula of calculating the area of an octagon is 2a ² (1+√2)

Here we know a = 4cm.

Now, if we put that in the equation,

2a² (1+√2) = 2 x 4² (1+√2)

= 2 x 16 (1+ √2)

≈77.25483 cm²

And the Perimeter of the Octagon will be- 8a

= 8 X 4cm

= 48cm.

Ques. Perimeter of an octagonal stop sign board is 16 cm. Find the area of the signboard? (3 marks)

Ans. We know, Perimeter of an Octagon = 8a

Here, 8a = 16cm. Hence, the a = 16/8= 2cm.

So if we put this in the formula of calculating area of an octagon, then it will be-

Area of an octagon = 2a ² (1+√2) = 2 x 2² (1+√2)

So, the area of the signboard is ≈ 19.31371 cm²

Ques. Perimeter of an octagonal stop sign board is 32 cm. Find the area of the signboard? (3 marks)

Ans. We know, Perimeter of an Octagon = 8a

Here, 8a = 16cm. Hence, the a = 32/8= 4cm.

So if we put this in the formula of calculating area of an octagon, then it will be-

Area of an octagon = 2a² (1+√2) = 2 x 4² (1+√2)

So, the area of the signboard is ≈ 77.25483 cm²

Ques.Calculate the area and perimeter of a regular octagon whose side is 6.5cm? (3 marks)

Ans. As we know the formula of calculating the area of an octagon is 2a ² (1+√2)

Here we know a = 6.5cm.

Now, if we put that in the equation,

2a ² (1+√2) = 2 x 6.52 (1+√2)

=204 cm²

And the Perimeter of the Octagon will be- 8a

= 8 X 6.5cm

= 52cm.

Ques. Find the length of each side of a regular octagon if the perimeter of the octagon is 240 units? (2 marks)

Ans. According to the regular octagon definition, all its sides are of equal length. The perimeter of an octagon = 8 × (side). Here, the perimeter is given as 160 units.

Thus, the length of each side = 240 ÷ 8 = 30 units.
Therefore, the length of each side of the octagon is 30 units.

Ques. A regular octagon is given which has a perimeter equal to 64cm. Find its area using the octagon formula? (3 marks)

Ans. Given: The perimeter of the octagon is 64 cm.

The perimeter of the octagon(P) = 8a
64 = 8a
a = 8cm
By using the octagon formula for the area
Area of octagon(A) = 2a²(1 + √2)
A = 2(8)²(1 + √2)
A = 308.96cm²

Ques. A regular octagon is given which has a perimeter of 48cm. Find its area using the octagon formula? (3 marks)

Ans. Given: The perimeter of the octagon is 72 cm.

The perimeter of the octagon(P) = 8a
72 = 8a
s = 9 cm
By using the octagon formula for the area
Area of octagon(A) = 2a²(1 + √2)
A = 2(9)²(1 + √2)
A = 229.068cm²

Ques. Find the length of each side of a regular octagon if the perimeter of the octagon is 400 units? (2 marks)

Ans. According to the regular octagon definition, all its sides are of equal length. The perimeter of an octagon = 8 × (side). Here, the perimeter is given as 160 units.

Thus, the length of each side = 400 ÷ 8 = 50 units.
Therefore, the length of each side of the octagon is 50 units.

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