Octagon

Types of Polygons

In Geometry, we study about different types of shapes. A polygon is any closed geometrical shape made up of straight lines that can be drawn on a flat surface, like a piece of paper. A circle or any other shape that includes a curve are not polygons.

Polygons are classified into different types depending on the number of sides they have:

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  1. Triangle (Polygon having 3 sides and 3 interior angles)

  2. Quadrilateral (Polygon having 4 sides and 4 interior angles)

  3. Pentagon (Polygon having 5 sides and 5 interior angles)

  4. Hexagon (Polygon having 6 sides and 6 interior angles)

  5. Heptagon (Polygon having 7 sides and 7 interior angles)

  6. Octagon (Polygon having 8 sides and 8 interior angles)

  7. Nonagon (Polygon having 9 sides and 9 interior angles)

  8. Decagon (Polygon having 10 sides and 10 interior angles)


In General, A Polygon with ‘n’ Sides Has: 

  1. ‘n’ interior angles. 

  2. Sum of interior angles = (n - 2) × 180°

  3. Each interior angle of a regular polygon = \[\frac{(n-2) х 180⁰}{n}\]

  4. Sum of exterior angles = 360°


 In this article, we will learn about the eight-sided polygon called “octagon” with its proper definition, shape, number of sides, properties, its formula of perimeter and area in detail.


Definition of Octagon

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An octagon is a polygon which has eight sides and eight angles. The word “octagon” is made up of two words, namely ‘octa’ and ‘Gonia’, which means eight angles. 

Since, octagon has 8 sides therefore,

  1. Sum of interior angles = (n - 2) × 180°

                                     = (8 - 2) × 180° = 6 × 180°

                                 = 1080°

  1. Each interior angle of a regular octagon = \[\frac{(n-2) х 180⁰}{n}\]

                                                                 = \[\frac{(8-2) х 180⁰}{5}\] = \[\frac{1080}{5}\]

                                                                 = 135°

  1. Sum of exterior angles = 360°


Shapes of Octagon

Depending on the sides, angles and vertices, octagon shapes are classified as:

  1. Regular octagons

  2. Irregular octagons


Regular Octagon

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To be a regular octagon the octagon must have:

  1. Eight congruent sides (sides of equal length)

  2. Eight congruent interior angles (each measuring 135°)

  3. Eight congruent exterior angles of 45°

Note: Regular octagons do not have parallel sides.

Irregular octagon

Irregular octagons are the octagon having different side lengths and angle measure.

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Irregular octagons can be a convex octagon or a concave octagon:

  • Convex octagon – An octagon having not any internal angles more than 180°.

  • Concave octagon – An octagon having one interior angle more than 180°.


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Properties of Octagon

  1. It has eight sides, eight vertices and eight interior angles.

  2. It has 20 diagonals.

  3. The sum of all interior angles is 1080°.

  4. The sum of the exterior angles is 360°.

  5. A regular octagon has all eight sides of equal length.

  6. Each interior angle of a regular octagon measures 135°.

  7. Irregular octagons have different side lengths and angle measures.

  8. All diagonals of the convex octagon lie inside the octagon.

  9. some diagonals of concave octagon may lie outside the octagon. 


The Perimeter of an Octagon

The perimeter of an octagon is the sum of the lengths of its eight sides.

For a regular octagon, since the length of all eight sides are equal. 

Therefore, the perimeter of a regular octagon = 8 × (side length) units.  


Area of an Octagon

Area of the octagon is the region covered by the sides of the octagon.

For a regular octagon, its area can be calculated by:

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  1. If the Measure of Side Length and Apothem is Given, Then:

(Apothem: a line from the centre of a regular polygon at right angles to any of its sides.)


Area of octagon = \[\frac{8}{2}\] × (side length) × (apothem) units2

OR, 

Area of octagon = \[\frac{1}{2}\] × (perimeter of octagon) × (apothem) units2


  1. If the Only Measure of Side Length is Given, Then:

Area of octagon = 2(1 + \[\sqrt 2\]) × (side length) 2 units2


  1. Length of the Longest Diagonal of an Octagon

If we join the opposite vertices of a regular octagon, then the diagonals formed have the length equal to:

 L = \[\sqrt {4 + 2√2}\] × (side length) units.

Longest diagonals are the axis of symmetry of octagon.


Solved Problems:

Q.1. Find the perimeter and area of a regular octagon whose side is 7 cm?

Solution: Given, side of octagon = 7 cm 

Perimeter of a regular octagon = 8 × (side length) units

                                                        = 8 × 7 cm

                                                        = 56 cm

And,

Area of octagon = 2(1 + \[\sqrt 2\]) × (side length) 2 units2

                                           = 4.828 × (7)2

                          = 236.572 cm2

Hence, the perimeter and area of a regular octagon whose side is 7 cm is 56 cm and 236.572 cm2 respectively.


Q.2. Find the length of the longest diagonal of a regular octagon whose side length is equal to 8 cm.

Solution:  Given, side of octagon = 8 cm 

 The length of the longest diagonal of a regular octagon = \[\sqrt {4 + 2√2}\] × (side length) units.

 L = \[\sqrt {4 + 2√2}\] × (side length)

 L = \[\sqrt {4 + 2√2}\] × 8 cm

L = 20.905 cm.