Octagon

In Geometry, we study 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 = \[\frac {(n-2)\times 180^0} {n}\]

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,

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

                                     = 1080°

\[\frac {(8-2)\times 180^0} {5}\] = \[\frac {1080} {5}\]

=135°

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 the 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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(Apothem: a line from the center 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

Area of octagon = 2(1 +2–√2) × (side length)² units²

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

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

Longest diagonals are the axis of symmetry of the octagon.

Solved Problems:

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 +2–√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.

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 =

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

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

 L = \[\sqrt{4+2\sqrt{2}}\] × 8 cm

L = 20.905 cm.

Octagon is a polygon that is studied in geometry; it has eight sides and eight angles. Various concepts related to an octagon like the area perimeter et cetera are studied throughout the schools that follow the curriculum set by the Central board of secondary education.

Students who want to get an in-depth analysis of an octagon can go through the notes provided by Vedantu, these notes are prepared by Vedantu’s expert research team, they have years of experience and are well versed in this field, Vedantu provides students with the latest material that is necessary to grasp in order to get a good score in the examination. Vedantu’s expert mathematics teachers have done a critical analysis of the various previous year question papers, this is done in order to help students get a clear perspective of what types of questions can be asked in the examination and how they should prepare for it. The concepts are written in an extremely simplified language so that students who may find it difficult to grasp the complex concepts that are covered in the NCERT book and go through the notes provided by Vedantu. These notes can give students an edge to perform well in the examinations as compared to other kids.

An octagon has eight vertices and eight edges, an octagon is considered as an eight-sided polygon also called 8-gon, in a two-dimensional plane. There are various types of octagons however, a regular octagon has all its sides equal in length. Each interior angle is equal to 135°, this gives us the measurement of all the exterior angles of an octagon which are 45° each. The combined sum of all the interior angles of the regular octagon is 1080°. It is a closed two-dimensional figure. If all the sides, interior angles and exterior angles of an octagon are equal then it is considered as a regular octagon otherwise it is known as an irregular octagon, the other types of octagons are convex and concave octagons.

Examples of an Octagon in Our Day to Day Lives Include –

A wall clock having eight edges, 

the stop sign boards at signals, 

The outline of an umbrella with eight ribs is also an octagon.

Properties of an Octagon –

The properties include mainly the properties of a regular octagon, they are as follows –

Regular octagons have eight sides and eight angles

The sum of all the exterior angles of an octagon is 360° where each angle measures 45°

The sum of all the interior angles of a regular octagon is 1080° very jungle measures 135°

A regular octagon contains 20 diagonals

All the sides and the angles of a regular octagon are equal.

Types of an Octagon-

There are four types of octagons – regular octagon, irregular octagon, concave, convex octagon.

We have already discussed the properties of a regular octagon.

An irregular octagon is an octagon that has unequal sides and unequal angles.

Convex Octagon- an octagon that has all its angles pointing outside and there are no angles pointing inwards is called a convex octagon. The angles of a convex octagon are less than 180°

Concave Octagon- an octagon that has one of its angles pointing inward is called a concave octagon.

Certain Concepts that are Related to the Study of Octagons are as Follows-

• Properties of the general octagon

• Regular octagon

• Area of an octagon

• Circumradius and inradius of an octagon

• Diagonals of an octagon

• Construction and elementary properties of an octagon

• Standard coordinates of an octagon

• Dissection of an octagon

• Skew octagon

• Petrie polygons

• Symmetry of octagon

• Uses of octagons

• Other uses

• Derived figures