Octagon Calculator

The regular octagon—an eight-sided polygon with equal sides and equal angles—is one of the most recognizable geometric shapes in everyday life. The stop sign, used in nearly every country in the world, is a regular octagon, chosen for its distinctive shape that is easily identified even when obscured by snow or damage. The Octagon Calculator computes all essential geometric properties of a regular octagon from the side length.

A regular octagon has interior angles of exactly 135° and exterior angles of 45°. Its area is given by the elegant formula A = 2(1 + √2)s², where s is the side length. The coefficient 2(1 + √2) ≈ 4.8284 means an octagon encloses nearly 4.83 times the square of its side length—significantly more than a square (which encloses s²) or a hexagon (which encloses ≈ 2.60s²). This high area-to-side ratio makes the octagon an efficient shape for enclosing space.

The formula A = 2(1 + √2)s² has an appealing geometric interpretation. A regular octagon can be constructed by cutting equal isosceles right triangles from the four corners of a square. If the octagon has side length s, the original square has side length s(1 + √2), and each corner triangle has legs of length s√2/2. The octagon's area equals the square's area minus the four triangles: [s(1+√2)]² − 4 × (1/2)(s√2/2)² = s²(1+√2)² − s² = 2(1+√2)s².

The apothem of a regular octagon—the perpendicular distance from the center to the midpoint of any side—is a = s(1 + √2)/2. This equals the inscribed circle radius and is used in the alternative area formula A = (1/2) × perimeter × apothem = 4s × s(1+√2)/2 = 2s²(1+√2), confirming consistency. The longest diagonal, connecting opposite vertices through the center, has length s√(4 + 2√2) and equals the diameter of the circumscribed circle.

Octagons appear frequently in architecture and design. The Dome of the Rock in Jerusalem, the Tower of the Winds in Athens, and many baptisteries and church floor plans employ octagonal geometry. In medieval Islamic architecture, the octagon served as a transitional shape between the square base and the circular dome, both structurally and symbolically. The octagonal floor plan combines the stability of the square with the spatial efficiency approaching that of a circle.

In modern engineering, octagonal cross-sections are used for columns, traffic bollards, and structural members where stress distribution benefits from the eight-fold symmetry. Octagonal tiles can create attractive floor patterns, though regular octagons alone cannot tile the plane—they require small square fillers at the vertices, producing the classic “truncated square” tessellation seen in many bathroom and kitchen floors.

This calculator provides instant results for area, perimeter, apothem, long diagonal, and interior angle, serving architects, engineers, students, and designers who work with octagonal geometry.