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 highway version is 30 inches across the flats (twice the apothem); smaller versions exist for local roads. Side length works out at 12.43 in and the area at 745.6 sq in, near enough 5.2 sq ft of sheet per sign.
• UFC fighting “octagon.” The eight-sided cage in mixed martial arts is 30 feet across the flats, the same geometry as a stop sign scaled from inches to feet, so the canvas comes to 745.6 sq ft. UFC quotes it as 750.
• 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.
• Deck framing. Octagonal decks and hot-tub surrounds are laid out from the across-flats dimension, then the joists are cut to the 22.5° angle at each corner.

Worked example: an 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, centre to mid-side; this is the “across flats / 2” measurement): r = (1 + √2) / 2 × s ≈ 1.207 × s
• Circumradius (centre to vertex): R = s × √(4 + 2√2) / 2 ≈ 1.307 × s
• Long diagonal (vertex to opposite vertex, through the centre): d_long = 2R = s × √(4 + 2√2) ≈ 2.613 × s
• Across-the-flats distance: 2r ≈ 2.414 × s
• Perimeter: P = 8s

Watch the circumradius. It is √(4 + 2√2)/2, not √(2 + √2)/2; the second version is a common typo and it produces 0.924s, which would put the corners inside the flats. Any octagon figure where the long diagonal comes out shorter than the across-flats distance has this error in it.

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 of side 1 has area 11.20, and the trend keeps going: fix the perimeter rather than the side, add sides, and the shape closes in on a circle, which is the figure that encloses the most area of all. An octagon with a 32 ft perimeter holds 77.3 sq ft; the circle with that same perimeter holds 81.5.

That last comparison is the practical one. Going from a square deck to an octagonal one of the same edge trim buys you real floor. Going from an octagon to a circle buys you 5% more and costs you every straight cut in the job, which is why nobody frames a round deck.