Octagon
Octagon is an eightsided twodimensional geometrical figure. An octagon consists of 8 interior angles and 8 exterior angles. The sum of the interior angles of an octagon is 1080°, and the sum of its exterior angles is 360°. There are 20 diagonals in an octagon. Octagons are classified into various types based upon their sides and angles. Let us learn more about the octagon shape in this article.
1.  What is an Octagon? 
2.  Types of Octagons 
3.  Properties of an Octagon 
4.  FAQs on Octagon 
What is an Octagon?
An octagon can be defined as a polygon with eight sides, eight interior angles, and eight vertices. When all the sides and angles of an octagon are equal in measurement, it is called a regular octagon. Every polygon is either convex or concave. Convex octagons bulge outwards, whereas concave octagons have indentations (a deep recess). Convex octagons are those in which all the angles point outwards. A regular octagon is an example of a convex octagon. The octagon in which at least one of its angles points inwards is a concave octagon.
Octagon Sides
An octagon is a polygon with 8 sides and 8 interior angles. The word 'Octagon' is derived from the Greek word, 'oktágōnon' which means eight angles. That is the reason why it is called an octagon.
Types of Octagons
Depending upon the sides and angles, an octagon is classified into the following categories:
 Regular and Irregular Octagon
 Concave and Convex Octagon
Regular Octagon
The octagon having eight congruent sides and angles is known as a regular octagon. A regular octagon has all the angles of equal measure.
 In a regular octagon, all the sides are equal in length, and all the angles are equal in measure.
 The interior angles add up to 1080° and the exterior angles add up to 360°.
 The interior angle at each vertex of a regular octagon is 135°.
Irregular Octagon
An octagon in which the sides and angles are not congruent is an irregular octagon. In other words, an irregular Octagon has eight unequal sides and eight unequal angles.
 It is an octagon with unequal sides and angles.
 All the interior angles are of different measure, but their sum is always 1080º.
Convex Octagon
The octagon, which has all its angles pointing outside and no angle pointing inwards, is convex. Each angle of a convex octagon is less than 180°.
 Convex octagons bulge outwards.
 None of their interior angles is greater than 180°.
Concave Octagon
The octagon in which one of the angles points inward is a concave octagon.
 Concave octagons have indentations (a deep recess).
 The interior angles are greater than 180°, that is, at least one angle is a reflex angle.
Properties of an Octagon
Here are a few properties of an octagon that can help to identify it easily.
 An octagon is a polygon with eight sides and eight angles.
 All its interior angles sum up to 1080°
 6 triangles can be formed in a regular octagon with the help of diagonals using a common vertex.
 It has 20 diagonals.
Octagon Diagonals
The diagonal of an octagon is the line segment that connects any two nonadjacent vertices. There are 20 diagonals in an octagon. The formula that is used to find the number of diagonals in any polygon is, Number of diagonals = n(n3)/2; where 'n' represents the number of sides of the polygon. In this case, there are 8 sides in an octagon. After substituting the value of 'n' in the formula, we get, Number of diagonals = n(n3)/2 = 8(8  3)/2 = (8 × 5)/2 = 20. Therefore, there are 20 diagonals in an octagon.
Angles of an Octagon
In a regular octagon, each interior angle is 135°. The sum of an octagon's interior angles is 1080°, and the sum of the exterior angles of an octagon is 360°.
 Sum of Interior angles = 135° × 8 sides = 1080°
 Sum of Exterior angles = 45° × 8 sides = 360°
Area of Octagon
The area of an octagon is the total space occupied by it. The formula to calculate the area of a regular octagon is, Area of a Regular Octagon = 2a^{2}(1 + √2); where 'a' is any one side length of the octagon. It is expressed in square units like inches^{2}, cm^{2, }and so on. For example, if one side of a regular octagon is 6 units, let us find the area of the octagon. Area of octagon = 2a^{2}(1 + √2), Substituting the value of 'a', Area of octagon = 2 × (6^{2}) × (1 + √2) = 72 × (1 + √2) = 173.8 square units.
In case of an irregular octagon, there is no specific formula to find its area. We divide the octagon into smaller figures like triangles. Then, after calculating the area of all the triangles, we add their areas to get the area of the octagon.
Perimeter of an Octagon
The perimeter of a polygon is the total length of its boundary. In order to calculate the perimeter of an octagon, the length of all the sides should be known. We know that in a regular octagon, all the sides are of equal length. Therefore, the formula that is used to find its perimeter is,
Perimeter of an octagon = Sum of all its sides
Perimeter of a regular octagon = 8a (Where 'a' is the length of one side of the octagon)
Important Notes
 An octagon has eight sides.
 The sum of all the interior angles in an octagon is always 1080º.
 The sum of all the exterior angles in an octagon is always 360º.
 A regular octagon has 20 diagonals.
 Regular octagons are always convex octagons, while irregular octagons can either be concave or convex.
Challenging Questions
 What is the sum of the number of sides, vertices, and angles in an irregular octagon?
a) 10 b) 8 c) 30 d) 40  What is the number of triangles that can be drawn in a regular octagon by connecting all the vertices?
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Octagon Examples

Example 1: How many triangles can be formed by joining the vertices of an octagon?
Solution:
In a regular octagon, by joining one vertex to the remaining vertices, 6 triangles can be formed.
However, if we consider all the vertices independently, we would have a total of 632 triangles.

Example 2: Find the length of each side of a regular octagon if the perimeter of the octagon is 160 units.
Solution:
According to the regular octagon definition, all its sides are of equal length. The perimeter of an octagon = 8 × (side). Here, the perimeter is given as 160 units. Thus, the length of each side = 160 ÷ 8 = 20 units. Therefore, the length of each side of the octagon is 20 units.

Example 3: Find the area of a regular octagon if its side measures 5 units.
Solution:
The sides of a regular octagon are of equal length. Here, the side length, a = 5 units. We can find the area of the octagon using the formula, Area of a Regular Octagon = 2a^{2}(1 + √2). Substituting the value of 'a' in the formula, we get, Area of a Regular Octagon = 2a^{2}(1 + √2) = 2 × (5)^{2} × (1 + √2) = 50 × (1 + √2) = 120.71 square units. Therefore, the area of the octagon is 120.71 square units.
FAQs on Octagon
What is an Octagon in Geometry?
Octagon is an eightsided twodimensional geometrical figure which consists of 8 interior angles and 8 exterior angles. The name 'octagon' is derived from the Greek word 'oktágōnon' which means eight angles.
What are the Angles of an Octagon?
There are 8 interior angles and 8 respective exterior angles in an octagon. In a regular octagon, all the interior angles are of equal measure and each interior angle measures 135°. Similarly, all the exterior angles are of equal measure and each exterior angle measures 45°. The sum of the interior angles of an octagon is 1080° and the sum of its exterior angles is 360°.
What is a Regular Octagon?
A regular octagon is one in which all the sides are of equal length and all the interior angles are of equal measure. The sum of its interior angles is 1080° and the sum of its exterior angles is 360°.
What is the Perimeter of an Octagon?
The perimeter of an octagon is the total length of its boundary. This can be calculated by adding the side lengths using the formula, Perimeter of octagon = Sum of all its sides. In case of a regular octagon, we use the formula, Perimeter of regular octagon = 8 × Side length, because all the sides are of equal length. The perimeter of an octagon is expressed in linear units like inches, cm, and so on.
What are the Properties of an Octagon?
The following properties of an octagon help us to identify it easily.
 An octagon has eight sides and eight angles.
 All its interior angles sum up to 1080°
 6 triangles can be formed in a regular octagon with the help of diagonals using a common vertex.
 It has 20 diagonals.
How to Find the Area of an Octagon?
The area of an octagon is the total space occupied by it. It is calculated with the formula, Area of a Regular Octagon = 2a^{2}(1 + √2); where 'a' is any one side length of the octagon. It is expressed in square units like inches^{2}, cm^{2}, and so on. In case of an irregular octagon, there is no specific formula to find its area. We divide the octagon into smaller figures like triangles. Then, after calculating the area of all the triangles, we add their areas to get the area of the octagon.
How many Triangles are there in an Octagon?
Using a common vertex, and with the help of diagonals, 6 triangles can be formed in an octagon.
How to find the Side Length of an Octagon?
The side length of an octagon can be calculated if the perimeter and the other sides are given. The sum of the given sides can be reduced from the perimeter to get the value of the unknown side. For example, if 7 sides of an octagon sum up to 36 units, and the perimeter of the octagon is 42 units, then the missing side = Perimeter  Sum of the remaining sides, which means, 42  36 = 6 units. In case of a regular octagon, the perimeter can be divided by 8 to get the value of one side of the octagon. For example, if the perimeter of a regular octagon is 96 units, then the length of one side = Perimeter ÷ 8 = 96/8 = 12 units.
How many Sides are there in an Octagon?
There are eight sides in an octagon. When all these eight sides are equal in length, it is known as a regular octagon, whereas when even at least one of the sides is different in measurement, it is known as an irregular octagon.
Is an Octagon a Quadrilateral?
No, an octagon is not a quadrilateral. A quadrilateral is a closed shape with four vertices and four sides and an octagon has 8 sides and 8 vertices.
What is the Angle Measure of the Angles that form the Corners of an Octagon?
The interior angle at each vertex of a regular octagon is 135° and each corresponding exterior angle is 45°.
Is any 8Sided Shape an Octagon?
If the shape is closed, made up of straight lines, and has eight sides, we call it an octagon.
Do Octagons have Equal Sides?
All octagons need not have equal sides. Octagons that have equal sides are known as regular octagons, while irregular octagons have different side lengths.
Where do the Longest Diagonal of an Octagon Meet?
An octagon has 20 diagonals in all and the longest ones are on its axes of symmetry, and they meet at the central point, which is also the origin of symmetry.
How many Parallel Lines does an Octagon have?
A regular octagon has 4 pairs of parallel sides. There are a total of 8 sides in an octagon, and those eight sides are parallel to their respective opposite side in the case of a regular octagon.
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