Octagon Calculator
Calculate area, perimeter, diagonals, and other properties of a regular octagon from its side length.
What is an octagon?
An octagon is an eight-sided polygon with eight angles. A regular octagon has all sides equal in length and all interior angles equal to 135deg. The sum of all interior angles in any octagon is 1080deg. Regular octagons appear in architecture (stop signs, gazebos), nature (crystals), and design. Irregular octagons have sides of different lengths. This calculator handles regular octagons, which have mathematical properties that allow calculation of area, perimeter, and other measurements from a single side length.
How do you calculate the area of an octagon?
For a regular octagon with side length a: Area = 2(1 + sqrt2) * a^2 ≈ 4.828 * a^2. This formula derives from dividing the octagon into triangles or using the apothem. Alternative formula: Area = 2a^2 * (1 + sqrt2), where a is the side length. For irregular octagons, divide into triangles and sum their areas. The regular octagon area formula is exact and used in engineering, construction, and mathematics. All measurements assume a regular octagon unless otherwise specified.
How do you find the perimeter of an octagon?
For a regular octagon: Perimeter = 8 * side length. This is straightforward because all eight sides are equal. For irregular octagons: add the lengths of all 8 individual sides. Example: Regular octagon with side length 5 cm has perimeter = 8 * 5 = 40 cm. Perimeter is used in fencing, framing, and material estimation. For real-world applications like building octagonal structures, always add extra material for waste and cuts.
What are other important octagon measurements?
Key octagon measurements: Inradius (apothem) = a * (1 + sqrt2)/2 ≈ 1.207a, the distance from center to midpoint of a side. Circumradius = a * sqrt(4 + 2sqrt2)/2 ≈ 1.307a, the distance from center to vertex. Diagonal (short) = a * sqrt(2 + sqrt2) ≈ 1.848a, connects vertices separated by one side. Diagonal (long) = a * (1 + sqrt2) ≈ 2.414a, connects opposite vertices. Interior angle = 135deg. These values are useful for construction, design, and geometric analysis.
What are real-world applications of octagons?
Octagons in the real world: Stop signs (universal traffic control), architectural features (gazebos, towers, buildings like Castel del Monte), umbrella shapes, clock faces, floor tiles, martial arts mats (MMA octagons), optical components. In construction: octagonal rooms maximize space efficiency. In engineering: octagonal shapes distribute stress well. Regular octagons tessellate with squares. Understanding octagon geometry helps in design, construction estimating, and spatial planning for these applications.