Octagon Area Calculator (regular)

A regular octagon has eight equal sides and eight 135° angles. From just the side length s, every measurement is fixed.

Area formula:

A = 2 × (1 + √2) × s² ≈ 4.828 × s²

A 10 cm regular octagon has area 482.84 cm². The 2(1 + √2) factor is irrational but easy to remember as roughly 4.83.

Where octagons show up in real life:

• Stop signs. Internationally standardized as red octagons. The US version is 30 inches across the flats (twice the apothem) for highway use; smaller versions exist for local roads. Area: about 558 sq in.
• UFC fighting “octagon.” The eight-sided cage in mixed-martial-arts has 30-foot flats. Floor area: ~676 sq ft.
• Gazebo and bandstand floors. Octagonal floors give 360° viewing without the building geometry getting awkward. A common gazebo size has 4 ft sides giving ~77 sq ft of floor space.
• Mansard roof corners on some Victorian architecture turn 90° via two 45° hips, creating octagonal floor plans for cupolas.
• Cookie and pastry molds are sometimes octagonal — close to round but easier to cut from a square sheet of dough.
• Many gazebos, gazebo decking, and outdoor decks use octagonal layouts for their balanced symmetry.

Worked example — octagonal gazebo with 4 ft sides:

A small backyard gazebo, regular octagon, 4 ft per side. Area = 4.828 × 16 = 77.25 sq ft of floor space.

That fits a small dining table and 4 chairs comfortably, or a hot tub plus a pair of lounge chairs.

Other useful measurements from the same side s:

• Apothem (inradius — center to mid-side; this is the “across flats / 2” measurement): r = (1 + √2) / 2 × s ≈ 1.207 × s
• Circumradius (center to vertex): R = s × √(2 + √2) / 2 ≈ 1.307 × s
• Long diagonal (vertex to opposite vertex, through center): d_long = 2R = s × √(2 + √2) ≈ 2.613 × s
• Across-the-flats distance: 2r ≈ 2.414 × s
• Perimeter: P = 8s

Sign-making rule of thumb. US stop signs are 30 inches across the flats, which means side length = 30 / (1 + √2) ≈ 12.43 inches. That awkward number is why stop signs are dimensioned by their across-flats measurement, not by side length.

Comparison to other shapes:

For the same side length s:

• Triangle (equilateral): area ≈ 0.433 × s²
• Square: area = 1.000 × s²
• Pentagon: area ≈ 1.720 × s²
• Hexagon: area ≈ 2.598 × s²
• Octagon: area ≈ 4.828 × s²

More sides means more area for the same edge length. A 12-sided polygon (dodecagon) of side 1 has area 11.20 — and as the number of sides goes to infinity, the polygon approaches a circle inscribed by that perimeter. The area-to-perimeter trend is exactly why circular shapes are so efficient.