Octagon is an eight-sided two-dimensional geometrical figure. An octagon consists of 8 interior angles and 8 exterior angles. The octagon has many interesting properties that make it a unique shape. The sum of its exterior angle is 360o and the sum of its interior angle is 1080o. There are 20 diagonals in an octagon.
In this article, we will learn the meaning and definitions of the octagon, its types, properties, area and perimeter of an octagon, and angles of an octagon, with some solved examples.
What is Octagon?
Octagon is a 2-D figure which consists of eight sides and eight angles. An octagon has 8 vertices and 8 edges. The word ‘octagon’ is derived from the Greek word ‘okta’ meaning eight and ‘gon’ meaning ‘side or angle’. Octagons are further classified as regular and irregular octagon. Every polygon is either convex or concave. A regular octagon is an example of convex octagon. Octagon shapes are generally used in stop signs, tiles, etc.
Octagonal Definition
Octagon is defined as an eight sided polygon also known as 8-gon. It has eight vertices, eight angles, and eight edges.
Octagonal Properties
Properties are generally considered for Regular octagon and its properties are given below :
- All 8 sides are equal and all 8 angles are also equal.
- There are total of 20 diagonals in a regular octagon.
- All 8 interior angles are equal, measuring 135o each and sum of all interior angle is always 1080o.
- All 8 exterior angles are equal, measuring 45o each and sum of all exterior angles is always 360o.
Octagon Angle
Octagon has eight sides and consist of eight angles. It has both interior and exterior angles. Angles inside the octagon are interior angles of octagon. Similarly, the angles at the exterior side of the octagon are termes as exterior angles. In an octagon, there are 8 interior and 8 exterior angles.
Interior Angles of Octagon
To find the measurement of each interior angle of any polygon, you can use formula
Interior angle = [(n-2) x 180] / n
For an octagon, n=8
Substituting this value into the formula:
Interior angle = [ ( 8-2 ) x 180 ] / 8
so, each interior angle of a regular octagon measure up to 135o.
The Sum of all the interior angle of an octagon, whether regular or irregular, is always 1080o.
Read More, Sum of Interior Angles of a Polygon
Exterior angle of an Octagon
To find the measurement of each exterior angle, you can use the formula:
Exterior angle = 360/n
For an octagon, n=8
Substituting this value into the formula :
Exterior angle = 360/8 = 45o
so, each exterior angle of a regular octagon measure up to 45o.
Types of Octagon
Depending on the sides, angles and vertices, an octagon is classified into the following categories:
- Regular and Irregular Octagon
- Concave and Convex Octagon
Regular Octagon : Octagon which has eight equal sides and eight equal angles, all the sides are equal in length, eight interior angle are measured to 135o and exterior angle of 45o each can be termed as regular octagon.
Irregular Octagon : Octagon which does not have all sides equal or all angles equal or we can also say whose sides and angles are not congruent is an irregular octagon. Even though its sides and angle are not equal but the sum of interior angles is always 1080o . Irregular octagon can be either convex or concave .
Concave Octagon : Octagon in which one of the angles is pointing inward and whose angle is greater than 180o are known as Concave Octagon.
Convex Octagon : Octagon which has all its interior angles less than 180o and all of its angles is pointing outward are called Convex Octagon.
Also Check, Convex Polygon
Octagon is widely used in geometry, few of its important formula are explained below:
- Perimeter
- Area
- Number of Diagonals
- Length of Diagonal
Octagon Perimeter
The total length of boundary of a Polygon is called as Perimeter of Polygon. To calculate the perimeter of octagon we must know length of all sides of octagon. We know that if the octagon is a regular octagon then knowing one side is enough as for regular octagon all sides and angles are equal. Therefore, the formula that is used for find Octagon perimeter is,
Perimeter of an Octagon = Sum of all its sides
Perimeter of a Regular octagon = 8 × s ( where s is length of one side of octagon )
Octagon Area
The area of an octagon is the total space occupied by it. The formula for calculating the area of Regular Octagon is :
Area of Regular Octagon = 2s2 × (1+√2)
where ‘s’ is the length of one side of octagon.
Note: For Irregular Octagon their is no such direct formula but we can calculate its area by subdividing the octagon into different shapes and then calculating the area of each sub divided shape and then summing up the areas of all the sub divided shapes.
Read More, Area of Octagon
Octagon Diagonal
The line segment between two non-adjacent vertices of octagon is diagonal of octagon. An octagon can consist of total 20 diagonals. The formula for calculating number of diagonal in a n-sided polygon contain given as : n × (n-3)/2.
Calculating the number of diagonals in Octagon using Formula
For Octagon, n=8
putting value of n into the formula, we get
n × (n-3)/2 = 8 × (8-3)/2 = 8 × 5/2 = 20
Therefore, an octagon can contain a total of 20 diagonals.
Length of the Diagonal of Octagon
The line segment between two non-adjacent vertices of octagon is diagonal of octagon. There are 3 types of diagonals in octagon based on their length :
- Short Diagonal
- Medium Diagonal
- Long Diagonal
Here, ‘s’ is the length of one side of regular octagon, c is short diagonal, a is medium diagonal, b is long diagonal.
Length of Octagon Diagonal Formula:
Short Diagonal ( c ) = s √( 2+√2 )
Medium Diagonal ( a ) = s ( 1+√2 )
Long Diagonal ( b ) = s √( 4+2 √2 )
Read More, Octagon Formula
Octagonal Line of Symmetry
Line of Symmetry is defined as a balanced proportionate similarity that is found in two halves of an object. Thus being an 8 sided polygon it has 8 line of symmetry. These line of symmetry can be drawn for an octagon and are drawn as given below :
Octagonal Prism
Octagonal Prism is a 3D ( Three Dimensional ) solid consist of two octagonal bases joined together by eight rectangular faces. It has total of 10 faces out of which 8 are rectangular and 2 are octagonal. It has 24 edges and 16 vertices. Altogether, the octagon prism has 10 faces, combining the surfaces of octagonal bases and rectangular faces.
Read More, Octagonal Prism.
Octagonal Pyramid
Octagonal Pyramid is a 3D figure with one octagon as the base with eight connecting triangles. It has total of 9 faces out of which 8 are triangle and 1 is octagon. It has 16 edges and 9 vertices. The vertex on top is the apex of the pyramid. The height of an octagonal pyramid is the length of line segment that’s perpendicular to the base which run through the apex of the pyramid.
Read More,
How to Draw an Octagon
To draw an octagon, plese use the following steps:
- Step 2. Draw two straight line perpendicular to each other that splits the circle into equal parts.
- Step 3. Now draw two Diagonal that intersect at the center and divide circle into equal parts.
- Step 4. Connect the tips of these spokes with straight lines.
- Step 5. Erase circle and all the lines inside the drawn figure, here you have a regular octagon.
Applications of Octagon
Some application or real-life based examples where we see octagon are :
- Stop sign : Stop sign are generally in shape of octagon with eight sided and eight angles.
- Clock : Clock are of different shapes in which their is octagon also.
- Medal : Medal as the sign of appreciation for his/her deed, some of the medal are of octagonal shapes.
- Trampoline : The frame of the trampoline is generally octagonal with eight sides, eight angles.
- Boxing Ring : The Boxing ring is a classic example of an octagon because of its eight sides.
Read More,
Solved Examples On Octagon
Example 1. Calculate the longest Diagonal of a Regular Octagon, whose Perimeter is 40cm ?
Solution:
Given that perimeter of Regular Octagon = 40cm. Therefore,Perimeter = 8 × s= 40 s = 40 /8s = 5 (length of each side of octagon)Now, length of longest Diagonal l = s √ (4+2√2) = 5 × 2.613 = 13.065 cm
Hence, the length of given diagonal is 13.065 cm.
Example 2. Find the Area of Regular Octagon whose side length is 10 cm ?
Solution:
Given, Length of side of the octagon = 10cm
Area = 2s2 × (1+√2)
= 2 × 100 × 2.414 = 482.8 cm2
Area of the given octagon is 482.8 cm2
Example 3. Find the Area of the regular octagon if its perimeter is 16 cm ?
Solution:
Given, Perimeter = 16 cm
8 × s = 16
s = 16/8
s = 2 ( length of side of Regular octagon )
Now, Area of Octagon= 2s2 × (1+√2)= 2 × 22 × (1+√2)= 2 × 4 × 2.414= 19.312 cm2
Example 4. Given The Length Short Diameter 20cm . Find the Area of Regular Octagon ?
Solution:
Given , Short Diagonal = s √( 2+√2 ) = 20cm, squaring both side ( s √( 2+√2 ) )2 = 400
s2 × (2+ √2) = 400
s2 = 400 / (2+ √2)
s = √ 117.157
s = 10.82 cm
Now, Area of Octagon = 2 × (1+√2)s2
Area = 2 × (1+√2) × (10.82)2
Area = 2 × 2.414 × 117.157
Area = 565.225 cm2
Hence, the area of the given octagon is 565.225 cm2
Practice Questions on Octagon
Q1. If the Side length of a regular octagon is 20cm. Calculate the perimeter of the octagon.
Q2. Two octagon A and B touching through one side and distance between their center is equal to the 40cm. If distance between their center is equal to longest diagonal length of A octagon. Find the area of the A octagon ?
Q3. Find the area of a octagon whose perimeter is 100cm.
Q4. Find the interior angle of a Octagon whose exterior angle is 82o.
Octagon Frequently Asked Questions
What are Properties of Regular Octagon ?
A regular octagon has 8 sides and 8 angles. Each interior angle are equal, measuring 135o each. Each Exterior angles are equal, measuring 45o each . It has 20 diagonals in all.
What is Sum of all angles of Irregular Octagon ?
The sum of all interior angles of a irregular polygon is always 1080o.
What are Classifications of Octagon?
Octagon are classified into following categories:
- Regular and Irregular Octagon
- Concave and Convex Octagon
What is difference Between Concave and Convex Octagon?
Octagon in which all of its interior angles is less than 180° and all angles are pointing outward are known as convex octagon whereas if any one of the angle in octagon has angle more than 180° is concave octagon.
What is Hexagon vs Octagon ?
Both Hexagon and Octagon are geometric shapes but differ in the number of sides. A Regular hexagon has 6 equal sides and 6 equal angles each angle of 120o whereas for regular octagon it has 8 equal side and 8 equal angles each angle of 135o.
What is Regular and Irregular Octagon ?
A regular octagon has all sides and all angles equal whereas in irregular octagon, all sides and angles are not equal.
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