Octagon – Definition – Shapes – Types – Formula – Properties
Octagon – Definition – Shapes – Types – Formula – Properties
What is an Octagon?
An octagon is a geometric shape that falls within the category of polygons, specifically an eight-sided polygon. The term “octagon” is derived from the Greek words “okto,” meaning eight, and “gonia,” meaning angles.
Below is a detailed exploration of octagons:
Definition
An octagon is defined as a polygon with eight sides and eight angles. Each of the sides connects at vertices (corners) to form eight interior angles.
![](http://mathsmd.com/wp-content/uploads/2024/06/image-7-258x300.png)
Types of Octagons
Octagons can be classified into two main types:
regular and irregular.
Octagons can be classified based on the equality of their sides and angles:
- Regular Octagon:
- Sides: All eight sides are of equal length.
- Angles: All interior angles are equal, each measuring 135°.
- Symmetry: It has symmetrical properties, including rotational and reflection symmetry.
- A regular octagon has 8 lines of symmetry.
- It also has rotational symmetry of order 8.
![](http://mathsmd.com/wp-content/uploads/2024/06/image-8-237x300.png)
Example: The stop sign is a common example of a regular octagon.
Irregular Octagon:
- Sides: The lengths of the sides are not equal.
- Angles: The interior angles are not necessarily equal.
- Symmetry: Lacks the symmetrical properties of a regular octagon.
Example: Many natural and man-made structures can form irregular octagons, like some floor tiles or decorative elements.
Formulas
1. Perimeter of a Regular Octagon
The perimeter (P) is the sum of all its sides. P=8a
where a is the length of one side.
2. Area of a Regular Octagon
The area (A) can be calculated using the following formula:
A=2(1+√2)a2
where a is the length of one side.
Alternatively, the area can also be calculated using the apothem (the perpendicular distance from the center to a side):
A=1/2×P×apothem
Properties of Octagons
1. Interior Angles:
- The sum of the interior angles of any octagon can be calculated using the formula: Sum of interior angles=(n−2)×180º
- For an octagon (n=8):
- Sum of interior angles=(8−2)×180∘ = 1080º
- In a regular octagon, each interior angle is: Each interior angle=1080/8=135º
2. Exterior Angles:
- The sum of the exterior angles of any polygon is always 360°.
- In a regular octagon, each exterior angle is: Each exterior angle=360/8=45º
3. Diagonals:
- The number of diagonals in an octagon can be calculated using the formula: Number of diagonals=n(n−3)/2
- For an octagon: Number of diagonals=8(8−3)2=20
- A regular octagon has 8 lines of symmetry.
- It also has rotational symmetry of order 8 (i.e., it looks the same after a rotation of 45º or any multiple of 45º.
Applications and Examples
- Architecture and Design: Octagonal shapes are often used in architecture, such as in buildings, towers, and decorative elements.
- Road Signs: The stop sign is a notable example of a regular octagon.
- Games and Puzzles: Many board games and puzzles use octagonal pieces or boards.
- Flooring: Octagonal tiles are common in flooring and other interior designs.
Summary
An octagon is a versatile and interesting polygon with eight sides and eight angles. Its properties, such as the sum of its interior angles, number of diagonals, and symmetrical characteristics, make it a fundamental shape in geometry. Understanding octagons provides a foundation for exploring more complex geometric shapes and their applications in various fields.