Octagon – Definition, Types, Formula, Properties, Examples

What is an Octagon ?

An octagon is a geometric shape that has eight sides and eight angles.

The word “octagon” comes from the Greek words “okto,” meaning eight, and “gonia,” meaning angle.

Here are some key characteristics of an octagon.

Definition:

An octagon is a polygon with eight sides and eight angles.

Types of Octagons

Regular Octagon:

All sides and all angles are equal. Each internal angle in a regular octagon measures 135 degrees.

Irregular Octagon:

The sides and angles are not allequal. The internal angles can vary, but the sum of the internal angles is always 1080 degrees.

Properties of a Regular Octagon

Sides and Angles:

All eight sides are of equal length, and each internal angle is 135 degrees.

Symmetry:

A regular octagon has 8 lines of symmetry and rotational symmetry of order 8 (it looks the same after a rotation of 45 degrees).

Diagonals:

The formula to find the number of diagonals in a polygon is = n(n−3)/2

where n is the number of sides.

For an octagon,

n = 8 so,

8(8−3)/2 = (8 x 5)/2=20

An octagon has 20 diagonals.

Area: The area A

Area of a regular octagon with side length

a can be calculated using the formula:

A=2(1+✓2)a²

Perimeter:

The perimeter P

P of a regular octagon is simply 8a,

where a is the length of one side.

Construction of a Regular Octagon

A regular octagon can be constructed by dividing a circle into eight equal parts using a compass and straightedge. Each point where the divisions intersect the circle is a vertex of the octagon. Connecting these vertices forms the octagon.

Special Formulas

Sum of Interior Angles:

Sum of interior angles=(n−2)×180º

For an octagon (n=8):

= (n−2)×180º

= (8−2)×180º

= 6×180∘=1080º

Sum of interior angles=6×180º=1080º

Each Interior Angle of a Regular Octagon:

Each interior angle=1080º/8

1080º/8=135º

=135º

Exterior Angles:

The exterior angle of a regular polygon can be calculated as:

360º/n

Each exterior angle=360º/n

For a regular octagon:

360º/8 = 45º

Each exterior angle=360º/8 = 45º

Summary

An octagon is an eight-sided polygon with significant geometric properties. A regular octagon, in particular, has equal sides and angles, making it symmetric and mathematically interesting. Understanding its properties, formulas, and construction methods provides comprehensive knowledge about this versatile and frequently encountered shape.