How to find the Area of an Octagon?

Geometry is a field of study of shapes and structures. It gives a brief explanation of the different shapes and their properties. Geometry gives defined formulas for the calculation of parameters of these flat or solid shapes. These formulas are different for each shape and are derived according to their dimensions. 

In the given article we have studied an eight-sided polygon viz. octagon along with its properties. The content of the article also gives the formula for the determination of the area of an octagon. Some sample numerical problems are included along with their solutions for reference.

Octagon

Octagon is a geometrical figure with 8 sides and 8 angles. The word octagon itself means “eight-sided”. An octagon is one of the plane figures or a polygon having eight sides. The interior angle of regular polygon measures to be 135 degrees each. And the exterior angles measure 45 degrees. All the midpoints of the sides of an octagon meet at its center and all the diagonals have the same length.

Octagon is a two-dimensional flat shape with eight sides and eight angles. It is a polygon made up of the joining of line segments. It has 8 sides and the sides are denoted by the letter ‘a’.

Octagon

Properties of an Octagon

• A regular polygon has eight sides.
• A polygon has eight equal angles.
• A regular polygon consists of 20 diagonals that meet at the center.
• Each interior angle measures to be 135°. And, the sum of all the interior angles equals 1080°.
• Each exterior angle measures to be 45°. And, the sum of all the exterior angles equals 360°.

Area of An Octagon

In geometry, there are set formulas for calculating the parameters of the shapes. The area of an octagon with its side length ‘a’ is given by the formula

 Area of an Octagon = 2a²(1 + √2)

Where,

a is the length of the side or edge

For Example:

If an octagon with a length of 8cm is given, its area can be calculated by

Area of an Octagon = 2a²(1 + √2)

A = 2(8)²(1 + √2)

A = 309.01cm²

The formula from the calculation of the area of an octagon can be derived by four different methods. These methods are briefly derived along with their diagrams.

• Method I

A regular octagon can be seen as a collection of eight small isosceles triangles sharing a common apex point. Hence, the area of a regular octagon can be calculated by determining the area of one of the triangles and multiplying it by 8.

Mathematically the area of the octagon is given by,

Area of octagon =  8 × Area of the triangle

We have been given an octagon with eight isosceles triangles. Consider one of the triangles from the octagon and draw a line perpendicular from its base to apex to form right angles.

Here, a is the length of the side of the octagon and OZ is the height of the triangle.

Now,

tan²θ = 1 – cos2θ/1 + cos2θ [SINCE, 2sin²θ = 1 – cos2θ and 2cos²θ = 1 + cos2θ]

tan²(45/2) = 1 – cos45°/1 + cos45°

tan²(45/2) = 1 – 1/√2/1 + 1/√2

tan(45/2) = √2 – 1

ZY/OZ = √2 – 1

OZ = a/2/√2 – 1

OZ = a/2 (1 + √2)

Area of triangle XOY =1 × XY × Oz

1/2 a × a/2 (1 + √2)

a²/4 (1 + √2)

Now, Area of octagon = 8 × area of triangle

Area of octagon = 8 × a²/4 (1 + √2)

Area of octagon = 2a²(1 + √2)

• Method II

When a regular octagon is divided into non-overlapping parts then, an octagon can be subdivided into a square, four rectangles, and four isosceles right-angled triangles.

Here, a is the length of the side of the octagon.

Now, the area of the square, A_sq = a²

Then, the area of the triangle = A_tr = 1/2 × x

Where,

x = √(a²/2)

Since, in a right-angled triangle, b² + h² = square of hypotenuse = side of octagon

Area of the rectangle, A_rec = x × a

Then the combined area of the given octagon will be,

Area of octagon = A_sq + 4 × A_rec + 4 × A_tr

• Method III

An octagon can be taken as a square with four triangles attached from each corner of the square.

hence, the side of the octagon ‘a’ with be the hypotenuse of the given triangle.

A² = 2x²

Let the length of the side of square will be 1 = a + 2x = a + 2√(a²/2)             

[Since, x = √(a²/2)]

The combined area of the octagon will be,

Area = (1 × 1) – 4 (1/2 x. x)

• Method IV

A regular octagon can also be conceptualized as a composition of 4 kites.

Let the diagonals of the kites be d and w and the area will be, 

Area of kite = d × w/2

Let us take the kite AHOB from the above diagram

∠HOB = 2π and HO = BO = r

And, HB = √2r

Since AO = r

Area of AHOB = AO × HB × 2

√(2r)²/2

Area of octagon = 4 × Area of kites

Area of octagon = 2 × √(2r)²

Irregular octagon

On contrary to a regular octagon an irregular octagon does not have sides and angles congruent to each other. Hence, an irregular octagon also has eight sides but is unequal with respect to each other.

The interior angles in an irregular octagon are always unequal but their sum always equals 1080° 

Area formula of an irregular octagon,

Like regular octagons, irregular octagons do not have the specific derived formula for the calculation of their area. So, to calculate the area of an irregular octagon it is divided into smaller figures like triangles, squares, and rectangles. and, later these all areas are added together.  

Sample Problems

Question 1: Find the area of a regular polygon with a side of 3cm.

Solution:

Given:

The side of the octagon is 3cm

Area of an Octagon = 2a²(1 + √2)

A = 2(3)²(1 + √2)

= 43.45cm²

Question 2: Find the area of a regular polygon with a side of 2.5cm. 

Solution:

Given:

The side of the octagon is 2.5cm 

Area of an Octagon = 2a²(1 + √2) 

A = 2(2.5)²(1 + √2) 

A = 30.17cm²

Question 3: Find the area of a regular polygon with a side of 7cm.

 Solution:

Given:

The side of the octagon is 7cm 

Area of an Octagon = 2a²(1 + √2)

 A = 2(7)²(1 + √2)

 A = 236.59cm²

Question 4: Find the area of a regular polygon with a side of 3.5cm. 

Solution:

Given, 

The side of the octagon is 3.5cm 

Area of an Octagon = 2a²(1 + √2) 

A = 2(3.5)²(1 + √2) 

A = 59.14cm²

Question 5: Find the area of a regular polygon with a side of 6cm.

Solution:

Given,

The side of the octagon is 6cm.

Area of an Octagon = 2a² (1 + √2)

A = 2(6)² (1 + √2) 

A = 173.82cm²

Question 6: Find the area of a regular polygon with a side of 5cm.

Solution:

Given,

The side of the octagon is 6cm.

Area of an Octagon = 2a² (1 + √2)

A = 2(5)²(1 + √2)

A = 120.71cm²

Question 7: Find the area of a regular polygon with a side of 10cm.

Solution:

Given,

The side of the octagon is 10cm.

Area of an Octagon = 2a² (1 + √2)

A = 2(10)² (1 + √2)

A = 482.84cm²