# 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 135Â°, summing up to 1080Â° degrees.

• Each of a regular octagonâ€™s exterior angle measures 45Â°, giving a total sum of 360Â°.

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 + âˆš2), where a represents the given octagonâ€™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 octagonâ€™s area. To calculate a triangleâ€™s area, draw a perpendicular line OP joining the apex to base ABâ€™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 = 90Â°Â

And angle OAP = 135/2Â°, and angle AOP = Î¸ = 45/2Â°, by construction.

To determine ODâ€™s length, solve trigonometric expressions for Î¸.

2 sinÂ² Î¸ = 1 - cos 2Î¸ â€“ (i)

2 cosÂ² Î¸ = 1 + cos 2Î¸ â€“ (ii)Â

Divide (i) by (ii),

tan2 Î¸ = (1 - cos 2Î¸) / (1 + cos 2Î¸) â€“ (iii)

Put the value of Î¸ in equation (iii),

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

Substitute the value of cos 45 as 1/âˆš2,

tan2 (45/2) = (1 â€“ 1/âˆš2) / (1 + 1/âˆš2) = (âˆš2 â€“ 1) / (âˆš2 + 1) = (âˆš2 â€“ 1)2/1

tan (45/2) = âˆš2 - 1

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

i.e., AP/OP= âˆš2 - 1

or, OP = AP / (âˆš2 â€“ 1) = (a/2) / (âˆš2 â€“ 1) = (a/2) (1+âˆš2)Â

Therefore, area of âˆ† OAB = (1/2) x AB x OP = (1/2) x a x (a/2) (1+âˆš2) = (a2/4) (1+âˆš2)

As discussed earlier, area of regular octagon= 8 x area of âˆ† OAB = 8 x (a2/4) (1+âˆš2)

Hence, proved the area of an octagon = 2a2 (1 + âˆš2).

## 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 polygonsâ€™ 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 sideâ€™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 octagonâ€™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 polygonâ€™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 âˆš( 4 + 2âˆš2), where a is the length of each side of the octagon.

The above formula can be derived as follows.

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In âˆ† AED in given diagram, AE2 + DE2 = AD2, (Pythagorasâ€™ theorem for right-angled triangle)

i.e., b2 + b2 = a2

â‡’ b = a / âˆš2 â€“ (iv)

Now, AB = 2b + a = aâˆš2 {substituting bâ€™s value form (iv)}, and BC = a.

In âˆ†ABC, AC2 = AB2 +BC2

Therefore, length of diagonal AC = a âˆš(4 + 2âˆš2).

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

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 180Â°, and pointed outwards.

Concave Octagon: These are octagons with any interior angle measuring beyond 180Â° 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 + âˆš2)

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 + âˆš2) + 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 + âˆš2), 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 + âˆš2).