Octagon Area Calculator
A regular octagon is an eight-sided polygon with all sides of equal length and all interior angles of 135 degrees. This calculator computes the area, perimeter, circumradius, and apothem of a regular octagon from the side length. The area formula equals 2 * (1 + sqrt(2)) times the square of the side length, or approximately 4.82843 times the side length squared. The perimeter is simply eight times the side length. The circumradius is the radius of a circle that passes through all eight vertices, and the apothem is the radius of a circle that touches the midpoint of all eight sides. Regular octagons are commonly seen in architecture and design, including the familiar octagonal stop sign.
Octagon formulas
Area = 2 * (1 + sqrt(2)) * s^2
Area (approximate) = 4.82843 * s^2
Perimeter = 8 * s
Circumradius (R) = s * sqrt(4 + 2*sqrt(2)) / 2
Circumradius (approximate) = 1.30656 * s
Apothem (a) = s * (1 + sqrt(2)) / 2
Apothem (approximate) = 1.20711 * s
Example calculation
For a regular octagon with side length 10:
- Area = 4.82843 * 10^2 = 482.84 square units
- Perimeter = 8 * 10 = 80 units
- Circumradius = 1.30656 * 10 = 13.07 units
- Apothem = 1.20711 * 10 = 12.07 units
Octagon area calculator: frequently asked questions
What is a regular octagon?
A regular octagon is an eight-sided polygon with all sides of equal length and all interior angles equal to 135 degrees. Octagons appear frequently in architecture and design, including stop signs and many building floor plans.
What is the formula for the area of a regular octagon?
The area of a regular octagon with side length s is: Area = 2 * (1 + sqrt(2)) * s^2, which simplifies to approximately 4.82843 * s^2. A regular octagon can be constructed by cutting the corners off a square.
What are the interior angles of a regular octagon?
Each interior angle of a regular octagon is 135 degrees. The sum of all eight interior angles is 1080 degrees, which follows the formula (n-2)*180 for an n-sided polygon where n = 8.
Where do regular octagons appear?
Regular octagons appear in stop signs, some architectural floor plans, and artistic designs. They are popular in urban planning and are often used for fountains, gazebos, and buildings due to their aesthetic appeal and practical tiling properties.
What are the circumradius and apothem of an octagon?
The circumradius of a regular octagon with side length s is approximately 1.30656 * s. The apothem is approximately 1.20711 * s. These are calculated using trigonometric functions with the central angle of 45 degrees.
Official sources
- Wolfram MathWorld: Regular Octagon.
- ISO 80000-2: Mathematical signs and symbols.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.