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.

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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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.

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

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.

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

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

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 ft^{2}.

Therefore, the area of a regular octagon is 173.8 ft^{2}.

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 m^{2}.

Therefore, the area of a regular octagon is 308.992 m^{2}.

Find the area for the given polygon:

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.

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

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

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.

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