Octagon Calculator

Calculate various properties of a regular octagon from its side length.

Enter the side length of the octagon to compute its area, perimeter, apothem (inradius), and circumradius.

Examples

Here are some examples to get you started.

Standard Octagon

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A standard octagon with a side length of 10 units.

Side Length: 10

Small Octagon

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A small octagon, for instance, used in a logo design, with a side length of 2.5 units.

Side Length: 2.5

Large Octagon for Architecture

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An octagon representing the base of a gazebo with a side length of 120 cm.

Side Length: 120

Fractional Side Length

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An octagon with a fractional side length to test precision.

Side Length: 7.75

Other Titles
Understanding the Octagon Calculator: A Comprehensive Guide
An in-depth look at the properties, formulas, and applications of regular octagons.

What is a Regular Octagon?

  • Definition of an Octagon
  • Properties of a Regular Octagon
  • Key Terminology
Definition of an Octagon
An octagon is a polygon with eight sides. A 'regular' octagon is one where all sides have equal length and all internal angles are equal (135°). This regularity makes its geometric properties predictable and calculable.
Properties of a Regular Octagon
A regular octagon has 8 equal sides, 8 equal interior angles of 135°, and 8 equal exterior angles of 45°. The sum of its interior angles is always 1080°. It possesses a high degree of symmetry, with 8 lines of reflectional symmetry and rotational symmetry of order 8.
Key Terminology
Side Length (a): The length of any one of the octagon's eight sides.
Perimeter (P): The total length of all sides combined (P = 8a).
Apothem (h) / Inradius (r): The distance from the center to the midpoint of a side. It is also the radius of the inscribed circle.
Circumradius (R): The distance from the center to any vertex. It is also the radius of the circumscribed circle.

Step-by-Step Guide to Using the Octagon Calculator

  • Entering the Side Length
  • Interpreting the Results
  • Using the Examples
Entering the Side Length
The calculator requires a single input: the side length of the regular octagon. Enter a positive number in the designated field. The calculator is designed to handle both integers and decimals.
Interpreting the Results
Once you click 'Calculate', the tool will display:
Area: The total space enclosed by the octagon.
Perimeter: The distance around the octagon.
Apothem (Inradius): The radius of the largest circle that can fit inside the octagon.
Circumradius: The radius of the circle that passes through all of the octagon's vertices.
Using the Examples
If you're unsure where to start, click on any of the provided examples. This will auto-fill the input field with a sample value, allowing you to see the calculator in action immediately.

Real-World Applications of Octagons

  • Architecture and Construction
  • Design and Art
  • Everyday Objects
Architecture and Construction
The octagonal shape is famously used in architecture. Gazebos, pavilions, and even entire buildings or rooms are designed with an octagonal footprint to create unique, panoramic views and interesting interior spaces. The famous Dome of the Rock in Jerusalem has an octagonal plan.
Design and Art
In design, octagons are used for everything from logos and patterns to decorative tiles and window frames. Its symmetry is aesthetically pleasing and can create a sense of balance and stability.
Everyday Objects
The most common example of an octagon in daily life is the stop sign. Its unique 8-sided shape makes it instantly recognizable to drivers. You can also find the shape in umbrellas, poker tables, and some nuts and bolts.

Practical Scenarios

  • Designing a gazebo with a side length of 2 meters.
  • Creating a tile pattern using octagons with 15 cm sides.
  • Calculating the material needed for a series of octagonal window frames.

Common Misconceptions and Correct Methods

  • Confusing Area and Perimeter
  • Irregular vs. Regular Octagons
  • Apothem vs. Circumradius
Confusing Area and Perimeter
A common mistake is to confuse area (the space inside) with perimeter (the distance around). The perimeter is a linear measurement (e.g., meters), while the area is a square measurement (e.g., square meters). Our calculator provides both values clearly to avoid confusion.
Irregular vs. Regular Octagons
This calculator is specifically for regular octagons. The formulas used do not apply to irregular octagons where sides and angles are not equal. Calculating the area of an irregular octagon is much more complex and typically requires dividing it into smaller, simpler shapes like triangles.
Apothem vs. Circumradius
It's easy to mix up the apothem and circumradius. Remember: the apothem goes from the center to the middle of a side, while the circumradius goes from the center to a vertex (corner). The circumradius will always be longer than the apothem.

Mathematical Derivation and Formulas

  • Area Formula
  • Perimeter Formula
  • Apothem and Radii Formulas
Area Formula
The area of a regular octagon can be calculated using the formula:
Area = 2 (1 + √2) a²
Where 'a' is the side length. This formula is derived by dividing the octagon into 8 isosceles triangles, finding the area of one triangle, and multiplying by 8.
Perimeter Formula
The perimeter is the most straightforward calculation:
Perimeter = 8 * a
Apothem and Radii Formulas
The apothem (h or inradius r) and circumradius (R) are calculated using trigonometry:
Apothem (h) = a / (2 tan(π/8)) = a (1 + √2) / 2
Circumradius (R) = a / (2 * sin(π/8))

Formula Examples

  • For a side 'a' = 5, Area = 2 * (1 + √2) * 5² ≈ 120.71
  • For a side 'a' = 5, Apothem = 5 * (1 + √2) / 2 ≈ 6.0355
  • For a side 'a' = 5, Circumradius = 5 / (2 * sin(22.5°)) ≈ 6.5328