 # Area of an Octagon

An octagon is a two-dimensional geometrical plane figure. In Geometry, we have studied different polygon shapes such as triangle, square, pentagon, hexagon, rectangle, etc. Like other shapes, the octagon is also a polygon. Octagon has 8 sides and 8 angles. It means the number of vertices and edges are 8. All the sides of the octagon joined with each other end-to-end to form a shape. These sides are in a straight line segment. They are not curved or disjoint with each other. ## Area of a Regular Octagon Formula

The area of an octagon is defined as the total space inside the boundary of an octagon. The measurement unit for the area is square units.

If a polygon is with eight equal sides and eight equal angles, then the polygon is a regular octagon. Otherwise, the polygon is known as an irregular polygon. The area of an octagon formula is given as,

Area of a regular octagon, A = 2a2 (1+√ 2 ) Square units.

Where “a” is the length of the octagon sides.

It is not possible to find the area of an irregular octagon using this formula. So, to find its area, it is divided into other regular polygons. Then, the areas of all polygons should be added to get its area.

### Area of an Octagon Examples

Question1: Find the area of regular octagon whose side is 5cm.

Solution:

Given:

Side, a = 5cm

We know that,

Area of a regular octagon, A = 2a2 (1+√ 2 ) Square units

A = 2(5)2 (1+√ 2 ) cm2

A = 2(25) (1+√ 2 ) cm2

A = 50 (1+√ 2 ) cm2

A = 50(1+1.414) cm2

A = 50(2.414) cm2

A = 120.7 cm2

Therefore, the area of a regular octagon is 120.7 cm2.

Question 2:

Find the area of an irregular octagon given below: Solution:

The given figure is an irregular octagon.

Therefore, the area of an irregular octagon ABCDEFGH is given below:

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

Finding the area of ABC:

Using Heron’s Formula:

S = (3+6+8)/2

S = 8.5

Therefore, the area of ABC = $\sqrt{S(S-a)(S-b)(S-c)}$ $A =\sqrt{8.5(8.5-3)(8.5-6)(8.5-8)}=7.64$

Therefore, the area of ABC = 7.64

Similarly, we can find the area of other triangles using Heron’s formula,

Area of ACD = 11.83

Area of ADE = 11. 39

Area of AFG = 27.81

Area of AGH = 35.99

Now, add all the areas of the triangle

Area of ABCDEFGH = 7.64 + 11.83 + 14.14 + 11. 39 + 27.81 + 35.99 = 108.83

Hence, the area of an irregular octagon is 108.83 square units