109 LearnersLast updated on July 11th, 2025
Volume of Octagon
The volume of an octagon refers to the theoretical space it would occupy if it were a solid shape. However, an octagon is typically a 2D polygon with eight sides, so it doesn't have volume in the traditional sense. In geometric contexts, we often discuss the area of an octagon. In this topic, let's explore the properties and area calculation of octagons, which can help in understanding concepts related to octagonal prisms or other 3D shapes with octagonal bases.
What is the volume of an octagon?
As a 2D polygon, an octagon does not have volume. Instead, it has an area and perimeter. If you are considering a 3D shape with an octagonal base, such as an octagonal prism, its volume can be calculated by multiplying the area of the octagonal base by the height of the prism.
The formula for the area of a regular octagon is: Area = 2(1 + √2) × side² Where 'side' is the length of one of the octagon's edges.
How to Derive the Area of an Octagon?
To derive the area of a regular octagon, we can divide it into simpler shapes, such as triangles. A regular octagon can be split into 8 isosceles triangles, each with a base equal to the side of the octagon.
Using trigonometry, the area of each triangle can be calculated and then multiplied by 8 to find the total area of the octagon.
The formula to calculate the area of a regular octagon is: Area = 2(1 + √2) × side²
How to find the area of an octagon?
The area of an octagon is always expressed in square units. To find the area, use the formula for a regular octagon: Area = 2(1 + √2) × side²
The side is the length of one edge of the octagon. This formula applies to regular octagons, where all sides and angles are equal.
Tips and Tricks for Calculating the Area of an Octagon
- Remember the formula: The formula for the area of a regular octagon is: Area = 2(1 + √2) × side²
- Break it down: The area is the space enclosed within the octagon. For a regular octagon, you just need to use the side length in the formula.
- Use approximations: If calculations become complex, approximate the square root of 2 as 1.414 for simpler arithmetic.
Common Mistakes and How to Avoid Them in Calculating the Area of Octagon
While learning about the area of octagons, some common mistakes might occur. Let’s look at some common errors and how to avoid them for a better understanding of octagonal geometry.
Mistake 1
Confusing Area with Perimeter
Some students confuse the formula for area with the formula for perimeter. The perimeter is calculated by 8 × side, while the area formula is 2(1 + √2) × side².
Mistake 2
Using the wrong formula for irregular octagons
The formula provided is for regular octagons only. If the octagon is irregular, the formula will not be accurate, and each triangle's area must be calculated separately.
Mistake 3
Incorrectly applying the area formula
Some might incorrectly apply the area formula by forgetting to square the side length or to multiply by 2(1 + √2), leading to incorrect results.
Mistake 4
Confusing linear and square measurements
Ensure you’re working with square measurements when calculating area, as opposed to linear measurements, which are used for perimeters or side lengths.
Mistake 5
Misapplying the square root of 2
When approximating √2, ensure the approximation is accurate to maintain precision in calculations.
Volume of Octagon Examples
Problem 1
An octagon has a side length of 4 cm. What is its area?
The area of the octagon is approximately 77.25 cm².
Explanation
To find the area of a regular octagon, use the formula: Area = 2(1 + √2) × side²
Here, the side length is 4 cm,
so: Area ≈ 2(1 + 1.414) × 4² = 2 × 2.414 × 16 ≈ 77.25 cm²
Problem 2
An octagon has a side length of 10 m. Find its area.
The area of the octagon is approximately 482.84 m².
Explanation
To find the area of a regular octagon, use the formula: Area = 2(1 + √2) × side²
Substitute the side length (10 m):
Area ≈ 2(1 + 1.414) × 10² ≈ 2 × 2.414 × 100 ≈ 482.84 m²
Problem 3
The area of a regular octagon is 125 cm². What is the side length?
The side length of the octagon is approximately 3.20 cm.
Explanation
If you know the area of the octagon and need to find the side length, rearrange the area formula:
side² = Area / 2(1 + √2) side ≈ √(125 / 2.828) ≈ 3.20 cm
Problem 4
An octagon has a side length of 2.5 inches. Find its area.
The area of the octagon is approximately 30.40 inches².
Explanation
Using the formula for the area: Area = 2(1 + √2) × side²
Substitute the side length 2.5 inches: Area ≈ 2(1 + 1.414) × 2.5² ≈ 2 × 2.414 × 6.25 ≈ 30.40 inches²
Problem 5
You have an octagonal garden with a side length of 3 feet. What is the area of the garden?
The garden has an area of approximately 43.45 square feet.
Explanation
Using the formula for the area: Area = 2(1 + √2) × side²
Substitute the side length 3 feet: Area ≈ 2(1 + 1.414) × 3² ≈ 2 × 2.414 × 9 ≈ 43.45 ft²
FAQs on the Area of Octagon
1.Is the area of an octagon the same as the perimeter?
2.How do you find the area if the side length is given?
3.What if I have the area and need to find the side length?
4.Can the side length be a decimal or fraction?
Important Glossaries for Area of Octagon
- Octagon: An eight-sided polygon with eight angles.
- Regular Octagon: An octagon with all sides and angles equal.
- Area: The space enclosed within a 2D shape, measured in square units.
- Perimeter: The total length of the boundary of a 2D shape.
- Square Units: The units of measurement used for area, such as cm², m², or in².
Seyed Ali Fathima S
About the Author
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.
Fun Fact
: She has songs for each table which helps her to remember the tables
