Octagon is a polygon in geometry, which has 8 sides and 8 angles. That means the number of vertices is 8 and the number of edges is 8. All the sides are joined with each other end-to-end to form a shape. These sides are in a straight line form; they are not curved or disjoint with each other. Each interior angle of a regular octagon is 135°. Therefore, the measure of exterior angle becomes 180° – 135° = 45°. The sum of the interior angles of the octagon is 135 × 8 = 1080°. In this article, let us discuss the octagon shape, its formulas, properties, and examples in detail.
Table of Contents:
- Octagon Shape
- Types of Octagon
- Regular and Irregular Octagon
- Convex and Concave Octagon
- Area of Regular Octagon
- Diagonal Length of Octagon
Shape of Octagon
Octagon is a geometrical shape in a two-dimensional plane. Like the other polygon shapes, we have studied in geometry, such as triangle, square, pentagon, hexagon, rectangle, etc., the octagon is also a polygon. The points which define it different from other geometrical shapes is that it has 8 sides and 8 angles.
If squares are built internally or externally on all the sides of an octagon, then the midpoints of the sections joining the centres of opposite squares form a quadrilateral: equi-diagonal and orthodiagonal ( whose diagonals length are equal and they bisect each other at 90 degrees).
You can see in the above figure, there are 8 sides of the polygon and have eight vertices as well. This is a regular octagon because all the angles and sides here are equal. In the same way, based on sides and angles, there are many types of polygons, such as:
- Decagon, and so on.
Types of Octagon
Depending upon the sides and angles of the octagon, it is classified into the following categories;
- Regular and Irregular Octagon
- Concave and Convex Octagon
Regular and Irregular Octagon
When an octagon has all equal sides and equal angles, then it is defined as a regular octagon. But if it has unequal sides and unequal angles, it is defined as an irregular octagon. See the figure below to see the difference between them.
A regular octagon is a closed shape with sides of the equal length and interior angles of the same measurement. It has eight symmetric lines and rotational equilibrium of order 8. The interior angle at each vertex of a regular octagon is 135°. The central angle is 45°.
In the above figure, the left-hand side figure depicts a regular octagon and the two figures on the right side shows irregular octagons. From the figure itself, we can analyse that there is a difference between the symmetry of regular and irregular polygons.
Convex and Concave Octagon
The octagon which has all its angles pointing outside or no angles are pointing inwards, is a convex octagon. The angles of the convex octagon are not more than 180°. And the octagon, which one of its angles pointing inward is a concave-shaped octagon.
In the above figure, you can see, the convex octagon has all its angles pointing outside from the center or origin point. Whereas on the right side, the concave octagon has one of the angles pointing towards inside the polygon.
Properties of Octagon
In the case of properties, we usually consider regular octagons.
- These have eight sides and eight angles.
- All the sides and all the angles are equal, respectively.
- There is a total of 20 diagonals in a regular octagon.
- The total sum of the interior angles is 1080°, where each angle is equal to 135°(135×8 = 1080)
- Sum of all the exterior angles of the octagon is 360°, and each angle is 45°(45×8=360).
Area of Regular Octagon
Area of the octagon is the region covered by the sides of the octagon. The formula for the area of a regular octagon which has 8 equal sides and all its interior angles are equal to 135°, is given by:
Perimeter of Octagon
The perimeter of the octagon is the length of the sides or boundaries of the octagon, which forms a closed shape.
Perimeter = Sum of all Sides = 8a
Length of the Diagonal of Octagon
If we join the opposite vertices of a regular octagon, then the diagonals formed have the length equal to:
L = a√(4 + 2√2)
Q.1: If the length of the side of a regular octagon is 5cm. Find its perimeter and area.
Solution: Given, a = 5cm
Therefore, Perimeter = 8a = 8 × 5 = 40cm
And Area = 2a2(1+√2) = 2 × 52 (1+√2) = 2 × 25 (1+√2)= 120.7 cm2
Q.2: If the side length of a regular octagon is 7cm. Find its area.
Solution: Given, length of the side of the octagon, a = 7cm
Area = 2a2(1+√2) = 2 (7)2(1+√2) = 236.6 sq.cm.
Q.3: Find the length of the longest diagonal of a regular octagon whose side length is equal to 10cm.
Solution: By the formula, we know, the length of the longest diagonal formula is given by:
L = a√(4 + 2√2)
L = 10√(4 + 2√2)
L = 10 x √6.828
L= 10 x 2.613
L = 26.13 cm
Q.4: Find the area and perimeter of a regular octagon whose side is of length 2.5 cm.
Solution: Area of octagon = 2a2(1+√2)
A = 2 x (2.5)2 (1+√2)
A = 12.5 x (1+√2)
A = 30.177 sq.cm
Perimeter of Octagon = 8 x sides of the octagon
P = 8 x 2.5
Frequently Asked Questions on Octagon
Define the shape octagon?
In geometry, an octagon is a polygon that has 8 sides and 8 angles. All the sides are joined end to end to form the shape octagon. The sum of the interior angles of an octagon is equal to 1080 degrees.
What are the different classifications of the octagon?
The octagon can be classified into:
Regular and irregular octagon
Convex and concave octagon
What is meant by the regular and irregular octagon?
If all the sides and angles of an octagon are of the same measure, then it is called the regular octagon. If the octagon has unequal side length and angle, then it is called an irregular octagon.
Define convex and concave octagon?
If all the angles of an octagon are pointing outwards and the interior angles are not more than 180 degrees, then it is called convex polygon. If anyone of the angles pointing inwards, then it is called a concave polygon.
Mention the properties of the octagon.
The important properties of the octagon are:
Octagon has 8 sides and 8 angles
The sum of the interior angles of an octagon is equal to 1080 degrees.
There are 20 diagonals in a regular octagon.
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