Area of Octagon Formula
Geometry is a branch of mathematics that delves into the study of shapes, their properties, and their dimensions. Among the myriad of geometric figures, octagons stand out as captivating and versatile polygons. In this comprehensive guide, we will embark on a journey to understand octagons, explore their various types, and master the methods and formulas to calculate their areas.
Unveiling the Octagon
An octagon is a polygon with eight sides, eight angles, and eight vertices. The term "octagon" is derived from two Greek words: "octa" (meaning eight) and "gonia" (meaning angle). This polygon comes in several forms, each with unique characteristics and applications.
Area of Octagon Formula
The area of an octagon, a polygon with eight sides, is a fundamental geometric concept. Understanding how to calculate the area of an octagon is essential for various applications in mathematics, engineering, and design. In this mini-lesson, we will delve into the details of the octagon area formula and explore how to calculate the area of this intriguing shape.
Breaking Down the Octagon
Before we dive into the formula, let's explore the key features of an octagon:
-
Eight Sides:
An octagon is a polygon with precisely eight sides, which are of equal or varying lengths.
-
Eight Angles:
Corresponding to its eight sides, an octagon also possesses eight angles. The interior angles of a regular octagon, where all sides and angles are equal, measure 135 degrees each.
-
Symmetry:
Regular octagons exhibit a high degree of symmetry, with multiple lines of symmetry that divide the shape into congruent sections.
Types of Octagons
Regular Octagon: In a regular octagon, all eight sides and angles are congruent, meaning they are equal in length and measure. Each interior angle of a regular octagon measures 135 degrees.
-
Irregular Octagon:
An irregular octagon has sides and angles of varying lengths and measures. It lacks congruency among its sides or angles.
-
Convex Octagon:
A convex octagon has interior angles that are all less than 180 degrees, meaning it does not fold inward.
-
Concave Octagon:
Conversely, a concave octagon has at least one interior angle that exceeds 180 degrees, causing the polygon to fold inward.
Also Check –
Perimeter of a Triangle Formula
Characteristics of a Regular Octagon
In a regular octagon, several intriguing properties come to light:
-
All sides have equal length, and all angles are congruent.
-
The sum of the interior angles of a regular octagon is always 1080 degrees.
-
The exterior angles of a regular octagon total 360 degrees.
-
Each interior angle measures 135 degrees.
Also Check –
Volume of a Hemisphere Formula
Calculating the Area of an Octagon
One of the fundamental aspects of geometry is determining the area enclosed by a polygon's sides. In the case of an octagon, there are various methods to calculate its area, depending on the information available. Below, we will explore three primary methods: using the side length, the apothem, and a combination of both.
-
Using the Side Length
What is the Area of Octagon?
An octagon, a two-dimensional polygon with eight sides, encompasses a specific amount of space within its boundaries, and this enclosed space is referred to as its area. The method to calculate the area of an octagon involves breaking it down into smaller isosceles triangles and computing their combined area. In the case of a regular octagon, where all sides have equal length and all angles are congruent, this process becomes more straightforward. Regular octagons exhibit interior angles measuring 135 degrees each, while their corresponding exterior angles measure 45 degrees each.
Also Check –
Volume of a Cylinder Formula
Area of Octagon Formula
The formula used to calculate the area of an octagon is:
2s
2
(1+√2),
where s is the length of the side of the octagon.
How to Calculate Area of an Octagon?
To determine the area of an octagon using the formula 2s²(1+√2), you can follow these steps:
Step 1:
Calculate the length of one side of the octagon.
Step 2:
Square the length obtained in Step 1.
Step 3:
Multiply the squared length from Step 2 by 2(1+√2).
Step 4:
Substitute the calculated value from Step 3 into the area of an octagon formula 2s²(1+√2).
Step 5:
Express the final result in square units.
These steps provide a systematic approach to finding the area of an octagon using the specified formula.
Also Check –
Volume of a Cube Formula
Area of Octagon Formula Examples
Example 1: Calculate the area of an octagon with a side length of 14 inches.
Solution:
Given side length, s = 14 inches.
Using the formula for the area of the octagon:
A = 2s
2
(1+√2)
A = 2 ×14
2
(1+√2)
A = 946.37
Therefore, the area of the octagon is approximately 946.37 square inches.
Example 2: Walter has an octagon with an area of 25.54 square units. Find the length of one of its sides.
Solution:
Given area, A = 25.54 square units.
Using the formula for the area of the octagon:
A = 2s
2
(1+√2)
25.54 = 2 × s
2
(1+√2)
s = 2.3 units
Therefore, the length of the side of the octagon is 2.3 units.
