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Area of an Octagon

The total space inside the boundary of an octagon is called as the area of an octagon. Area is measured in terms of square units.

Area of an Octagon

In the figure shown above, the octagon is a regular octagon. This is because, the sides and angles are of the same measure. Area of this octagon can be found easily with the help of the formula. But, the area of an irregular formula cannot be found with the help of any particular formula.

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Area of an Octagon Formula

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The total space inside the boundary of an octagon is called as the area of an octagon. Area is measured in terms of square units.

The formula for area of a regular octagon is given as follows,

Area of a octagon = $\text{side}^2 \times (2 + 2(\sqrt{2}))$.

For an irregular octagon, it is not possible to find it's area with the help of a single formula. So, it has divided in to other regular polygons and then, the areas of all polygons should be added to get its area.

Area of a Regular Octagon

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If in a polygon, all eight sides are equal and eight angles are equal, then this polygon is a regular octagon.

Area of a Regular Octagon Formula:


The area of an regular octagon formula is as follows,

Area = ( 2 + 2 $\sqrt{2}$ )$a^2$ = 2(1 + $\sqrt{2}$)$a^2$

where, a is the length of the side.

Alternate Method:

The area of a regular polygon of n sides and radius r is given by n$r^2$ sin($\frac{\pi}{n}$) cos($\frac{\pi}{n}$).

How to Calculate Area of Octagon?

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Let us see with the help of some examples how to calculate area of an octagon.

Solved Examples

Question 1: Find the area of a regular octagon whose side is 6 ft.
Solution:

Given that, side s = 6 ft.

We know that the area of a regular octagon = $\text{side}^2 \times (2 + 2(\sqrt{2}))$

= $6^2 \times (2 + 2(1.414))$

= $36 \times (2 + 2.828)$

= $36 \times (4.828)$

= 173.8 ft2.

Therefore, the area of a regular octagon is 173.8 ft2.



Question 2: Find the area of a regular octagon whose side is 8m.
Solution:

Given that, side s = 8 m.

We know that the area of a regular octagon = $\text{side}^2 \times (2 + 2(\sqrt{2}))$

= $8^2 \times (2 + 2(1.414))$

= $64 \times (2 + 2.828)$

= $64 \times (4.828)$

= 308.992 m2.

Therefore, the area of a regular octagon is 308.992 m2.



Question 3:

Find the area for the given polygon:


Calculate Area of Octagon
Solution:
Given polygon is an irregular octagon.

Area of ABCDEFGH = Area of ABC + Area of ACD + Area of ADE + Area of AEF + Area of AFG + Area of AGH.

Step 1: Area of ABC

Here, AB = 3, BC = 6 and AC = 8

S = $\frac{3+6+8}{2}$ = 8.5

Area = $\sqrt{S(S - AB)(S - BC)(S - AC)}$ =
$\sqrt{8.5 * 5.5  * 2.5 * 0.5 }$ = $\sqrt{58.44}$ = 7.64

Step 2: Similarly, by finding the area of other triangles by Heron's formula, we get

Area of ACD  = 11.83

Area of ADE = 14.14

Area of AEF = 11.39

Area of AFG = 27.81

Area of AGH = 35.99

Step 3:

Area of ABCDEFGH = 7.64 + 11.83 + 14.14 + 11.39 + 27.81 + 35. 99 = 108.83 sq units.

Practice Problems on Area of an Octagon

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Given below are some of the practice problems on area of an octagon.

Practice Problems

Question 1: Find the area of a regular octagon whose side is 78 ft.
Question 2: Find the area of a regular octagon whose side is 100 m.
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