What is a Polygon?
A polygon is a closed shape made by line segments. A polygon is made of two terms “poly” and “gon”. The term “poly” means many, and “gon” means angles. Thus, a polygon contains many angles. The smallest polygon is a triangle, a closed shape with three sides.
What is Octagon?
An octagon is a polygon containing eight sides. Thus, an octagon has eight edges, eight angles and eight vertices. When all the sides and angles of an octagon are equal then it is called a regular octagon.
Regular Octagon
Angles in an Octagon
There are two types of angles in a polygon, interior angles and exterior angles. In an octagon there are eight interior and eight corresponding exterior angles. The sum of all interior angles of an octagon is 1080°, and the sum of all the exterior angles of any polygon is 360°.
Classification of Octagons
An octagon is classified on the basis of side length and angle measure.
When we consider the sides, there are two types of octagons:
- Regular octagon: When all sides of an octagon are equal, then, it is called a regular octagon.
- Irregular octagon: When any two sides of an octagon differ in length, then, it is called an irregular octagon.
When considering the angle measurements, there are two types of octagons:
- Convex octagon: When all the interior angles of an octagon are less than 180°, then, it is called a convex octagon.
- Concave octagon: If any of the interior angles exceed 180°, then it is a concave octagon.
Perimeter of an Octagon
As we know that perimeter is the length of the boundary of a geometric shape, in the case of an octagon, it has 8 sides as a boundary. So, the sum of all the sides of an octagon gives us its perimeter.
Perimeter of octagon P = sum of the length of all sides
We know that all the sides of a regular octagon are equal,
Hence, the perimeter of a regular octagon, P = 8 \(\times\) side
Rapid Recall
- Octagons have eight sides and eight angles.
- All the sides and all the angles are equal in the regular octagon.
- The sum of all interior angles is 1080°, and each angle of the regular octagon is 135°.
- Sum of all exterior angles of an octagon is 360° and each exterior angle of a regular octagon is 45°.
- Perimeter of regular octagon (P) = 8 \(\times\) side.
Solved Examples
Example 1: Find the perimeter of a regular octagon if the length of one side is 7 inch.
Solution:
P = 8 \(\times\) side Write the formula for perimeter
P = 8 \(\times\) 7 Substitute 7 for side
P = 56
So, the perimeter of a regular octagon is 56 inches.
Example 2: If the perimeter of a regular octagon is 168 feet Find the side length.
Solution:
P = 8 \(\times\) side Write the formula for perimeter
168 = 8 \(\times\) side Substitute 168 for P
\(\frac{168}{8}~=~\frac{8}{8}~\times\) side divide each side by 8
21 = side Simplify
So, the side length of a regular octagon is 21 feet.
Example 3: Name the octagon, if two interior angles of an octagon are 130° and 260°.
Solution: Since, 260° is greater than 180°.
So, the given octagon is a concave octagon.
There are eight vertices in an octagon.
The measure of each interior angle of a regular octagon is 135°.
On the basis of side length: Regular and irregular octagon
On the basis of angle measure: convex and concave octagon.
The measure of each exterior angle of a regular octagon is 45°.