Area of an Octagon

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Any 2-dimensional shape with more than 2 line segments is called a polygon. Some known polygon examples include square, triangle, rectangle, pentagon, etc. Likewise, polygons with 8 sides in a 2-dimensional plane are known as an octagon. These shapes can again be divided into categories depending on their side and angle measurements. In this section, students will learn about the different types of octagons and how that affects their properties and calculations, including the area of an octagon.

Read on to get acquainted with the basics before heading on with the formulae.

Types of Octagons

Octagons can be divided into 2 types, based on their side lengths.

  1. Regular Octagon:

An octagon with equal-length sides and all the same value angles is called a regular octagon.

  1. Irregular Octagon:

If octagon sides and angles are unequal, it is called an irregular octagon.

Properties of a Regular Octagon

Following are the features of a regular octagon.

  • It comprises 8 sides and 8 angles.

  • The measurement of each side is equal, as is the case with its angles.

  • A regular octagon comprises a total of 20 diagonals.

  • Each interior angles measures 135°, summing up to 1080° degrees.

  • Each of a regular octagon’s exterior angle measures 45°, giving a total sum of 360°.

Based on the above characteristics, students can derive many other properties of an octagon, which they will need to solve the varied numerical that they will encounter under this topic. Read on to know how to derive some of these like area, perimeter, and diagonal formula for an octagon.

Area of a Regular Octagon

The most widely used formula to calculate the area of regular octagon is given as:

A = 2a2 (1 + √2), where a represents the given octagon’s each side length.

To derive this equation, consider the given pentagon. Drawing all its diagonals has divided it into 8 isosceles triangles with the centre as their common apex.

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Calculating the area of one of these similar triangles, and multiplying it by 8 will give you the given octagon’s area. To calculate a triangle’s area, draw a perpendicular line OP joining the apex to base AB’s bisectional point, as shown in the figure below. This is an apothem of octagon, which can be calculated by dividing any of the longest diagonals by 2.

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Here, AB = side of given octagon = a

Therefore AP = PB = a/2

Now, angle OPB = angle OPA = 90° 

And angle OAP = 135/2°, and angle AOP = θ = 45/2°, by construction.

To determine OD’s length, solve trigonometric expressions for θ.

2 sin² θ = 1 - cos 2θ – (i)

2 cos² θ = 1 + cos 2θ – (ii) 

Divide (i) by (ii),

tan2 θ = (1 - cos 2θ) / (1 + cos 2θ) – (iii)

Put the value of θ in equation (iii),

tan2 (45/2) = (1 - cos 45) / (1 + cos 45)

Substitute the value of cos 45 as 1/√2,

tan2 (45/2) = (1 – 1/√2) / (1 + 1/√2) = (√2 – 1) / (√2 + 1) = (√2 – 1)2/1

tan (45/2) = √2 - 1

Since angle AOP = 45/2°, tan (45/2) = AP/OP

i.e., AP/OP= √2 - 1

or, OP = AP / (√2 – 1) = (a/2) / (√2 – 1) = (a/2) (1+√2) 

Therefore, area of ∆ OAB = (1/2) x AB x OP = (1/2) x a x (a/2) (1+√2) = (a2/4) (1+√2)

As discussed earlier, area of regular octagon= 8 x area of ∆ OAB = 8 x (a2/4) (1+√2)

Hence, proved the area of an octagon = 2a2 (1 + √2).

Area of an Irregular Octagon

The above formulae to calculate the area of an octagon were for the condition that its every side and angle is equal. For irregular shapes like this one given below, the previous formulae are obsolete.

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In this scenario, to calculate the area of an octagon formula, you need to visualise a given octagon divided into different polygons, whose necessary measurements are known or can be derived from given information. Now, calculate each of these polygons’ area with their known area formulae. Lastly, add calculated areas of these polygons to find the area of the octagon.

The formula to find the area of some common polygons, besides triangle are given below.

  • Area of square = s2, where s represents each side’s length.

  • Area of rectangle = l x b, where l stands for length and b represents width.

  • Area of parallelogram = b x h, where b is its base length, and h is its height.

Perimeter of an Octagon

The total distance covered by an octagon’s periphery is called its perimeter. In other words, it is the length of its boundary. Therefore calculating the perimeter of the octagon formula is nothing but a sum of the measurement of all its sides. Therefore, it can be represented as:

Perimeter of octagon = 8a, where a is the given polygon’s each side length.

Length of a Diagonal of an Octagon

The diagonal of an octagon is a line drawn to join any pair of its opposite vertices. A regular octagon comprises 4 equal diagonals, each of which divides it into two similar pentagons. The formula to calculate their length is given as:

L = a √( 4 + 2√2), where a is the length of each side of the octagon.

The above formula can be derived as follows.

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In ∆ AED in given diagram, AE2 + DE2 = AD2, (Pythagoras’ theorem for right-angled triangle)

i.e., b2 + b2 = a2

⇒ b = a / √2 – (iv)

Now, AB = 2b + a = a√2 {substituting b’s value form (iv)}, and BC = a.

In ∆ABC, AC2 = AB2 +BC2

Therefore, length of diagonal AC = a √(4 + 2√2).

Now, you can implement diagonal length calculated by this formula to find the area of an octagon. 

The Area of octagon for class 8 is an important concept to help students develop a clear foundation of geometry for efficient numerical solving as well as higher studies. A thorough study of prescribed textbooks undoubtedly comes first for exam preparation. For a more detailed understanding of this topic along with illustrations and exercises on different types of problem sums, refer to online tutoring sites like Vedantu. Download the app to avail live expert guidance.

FAQ (Frequently Asked Questions)

1. What are Concave and Convex Octagons?

Ans: Based on angle measurements, octagons are divided into 2 types:

Convex Octagon: An octagon with all interior angles less than 180°, and pointed outwards.

Concave Octagon: These are octagons with any interior angle measuring beyond 180° and pointed inwards.

2. How to Calculate the Surface Area of an Octagonal Prism?

Ans: To find the surface area of a regular octagonal prism, students need to calculate area of octagonal surface and area of the sides:

Area of 2 octagonal surfaces = 2. 2a2(1 + √2)

Area of 8 sides = 8. a

Where a is the length of each edge.

Add these 2 up to get the total surface area.

Hence, total surface area of octagonal prism = 4a2(1 + √2) + 8a.

3. How to Derive Octagon Volume Formula?

Ans: To calculate the volume of a regular 3-dimensional octagon, you will first need the formula for area of its octagonal surface. Derive the area formula as already discussed to get area A = 2a2(1 + √2), where a = length of each edge.

Multiply this area with height of given octagonal prism h, to get its volume.

Therefore, volume of an octagon = h. 2a2(1 + √2).