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Last updated on September 17, 2025
Area is the space inside the boundaries of a two-dimensional shape or surface. There are different formulas for finding the area of various shapes/figures. These are widely used in architecture and design. In this section, we will find the area of the octagon.
An octagon is a polygon with eight sides and eight angles. A regular octagon has all sides of equal length and all interior angles are equal. The area of the octagon is the total space it encloses.
To find the area of a regular octagon, we use the formula: Area = 2 × (1 + √2) × a², where 'a' is the length of a side. Let's understand how this formula is derived.
Derivation of the formula: A regular octagon can be divided into 8 isosceles triangles by drawing lines from the center to each vertex. For one triangle, the base is the side 'a' of the octagon, and the height can be found using trigonometry. The area of one triangle is (1/2) × base × height. The total area of the octagon is 8 times the area of one triangle, which simplifies to 2 × (1 + √2) × a².
The area of a regular octagon can be found using the side length. Here’s the method: Method Using the Side Length If the side length 'a' is given, use the formula Area = 2 × (1 + √2) × a².
For example, if the side length is 5 cm, what will be the area of the octagon? Area = 2 × (1 + √2) × 5² = 2 × (1 + √2) × 25 = 2 × (1 + 1.414) × 25 = 2 × 2.414 × 25 = 120.7 cm² The area of the octagon is approximately 120.7 cm².
We measure the area of an octagon in square units. The measurement depends on the system used: In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²).
In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²).
A regular octagon has equal sides and equal angles, making the formula for its area consistent. However, for irregular octagons, different methods may be required:
Case 1: Regular Octagon Use the formula Area = 2 × (1 + √2) × a², where 'a' is the side length.
Case 2: Irregular Octagon An irregular octagon requires dividing it into known shapes, such as triangles or rectangles, and summing their areas.
To ensure correct results while calculating the area of an octagon, consider these tips and tricks:
It is common for individuals to make mistakes while finding the area of an octagon. Here are some common mistakes:
The side length of a regular octagon-shaped tile is given as 12 cm. What will be the area?
The area is approximately 695.4 cm².
Here, the side length 'a' is 12 cm.
The area of the octagon = 2 × (1 + √2) × a² = 2 × (1 + √2) × 12² = 2 × 2.414 × 144 = 695.4 cm².
What will be the area of a regular octagon if the side length is 7 m?
The area is approximately 284.6 m².
If the side length is 7 m, the formula Area = 2 × (1 + √2) × a² is used.
So, the area = 2 × (1 + √2) × 7² = 2 × 2.414 × 49 = 284.6 m².
Calculate the area of a regular octagon with a side length of 10 cm.
The area is approximately 482.8 cm².
To find the area, use the formula Area = 2 × (1 + √2) × a².
Here, a is 10 cm. So, Area = 2 × (1 + √2) × 10² = 2 × 2.414 × 100 = 482.8 cm².
Find the area of a regular octagon with a side length of 15 inches.
The area is approximately 1309.6 in².
Using the side length of 15 in, the area is calculated as Area = 2 × (1 + √2) × 15² = 2 × 2.414 × 225 = 1309.6 in².
Help Lisa find the area of her regular octagon-shaped garden if each side is 8 m long.
The area is approximately 309.0 m².
The side length is 8 m.
Use the formula Area = 2 × (1 + √2) × a² = 2 × 2.414 × 64 = 309.0 m².
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
: She has songs for each table which helps her to remember the tables