The octagon calculator above solves a regular eight-sided figure from whichever dimension you have to hand. Octagons are almost never specified by their side length in practice — a gazebo is quoted by its overall width, a sign by its face size, a table top by its diameter — so the tool converts from any of five starting measurements rather than forcing you to work backwards first.
Arb Digital publishes this as part of a free set of plane-geometry tools. The section on cutting an octagon from a square is the part most people come here for, because it turns the maths into a single mark-out number you can take to a workbench.
What This Octagon Calculator Does
It reports the area, perimeter, apothem, circumradius, across-flats width and across-corners width of a regular octagon: eight equal sides, eight equal 135° angles. Enter a 10 cm side and you get an area of about 482.84 cm², a perimeter of 80 cm, an apothem of 12.071 cm, an across-flats width of 24.142 cm and an across-corners width of 26.131 cm.
The quantity field scales the area across multiple identical octagons. As with all the regular-polygon tools on this site, it assumes the shape is regular. An eight-sided outline with unequal edges is not covered by any single formula and needs corner coordinates — the polygon calculator takes those directly.
How to Use It
- Work out which dimension you have. A quoted "diameter" for an octagonal table usually means across corners; a quoted width for a sign or a paving slab usually means across flats.
- Choose that measurement type in the dropdown so the conversion to side length is done correctly.
- Enter the value and select a unit. The area comes back in that unit squared.
- Add a quantity if you need a total across several identical pieces.
- Read the working line to see the conversion and the area formula applied to your figures.
The Formula: How Octagon Area Is Calculated
For a regular octagon of side s, the area is A = 2(1 + √2)s², approximately 4.828427 × s². With s = 10 cm, A = 4.828427 × 100 = 482.84 cm². The 1 + √2 term is the silver ratio, and it appears here for the same reason √3 appears in the hexagon: it is the trigonometric constant for the internal angle, since the general regular-polygon area is (n/4)s² cot(π/n), and cot(π/8) = 1 + √2.
The universal cross-check applies here too: A = ½ × perimeter × apothem. With a perimeter of 80 cm and an apothem of 12.0711 cm, that is ½ × 80 × 12.0711 = 482.84 cm², which agrees exactly. The apothem is s(1 + √2)/2 ≈ 1.207107s, and the circumradius is s√(2 + √2)/2 ≈ 1.306563s. MathWorld's octagon entry derives these relations formally.
Cutting a Regular Octagon From a Square
This is the most useful practical result on the page. Take a square of side W and cut a right-angled triangle off each corner. If the cuts are the right size, what remains is a regular octagon whose across-flats width is exactly W. The mark-out distance from each corner along each edge is W(2 − √2)/2 ≈ 0.292893 × W.
Work it through on a 1,200 mm square panel. Measure 0.292893 × 1,200 ≈ 351.5 mm in from every corner along both edges, join those marks, and cut. The resulting octagon has sides of 1,200 − 2 × 351.5 = 497 mm, which matches W(√2 − 1) = 1,200 × 0.414214 ≈ 497 mm as expected. Its area is 4.828427 × 497² ≈ 1,192,600 mm², or about 1.193 m².
Compare that to the square you started with: 1.44 m². The octagon retains 82.84% of the square's area, meaning the four corner triangles account for just over 17%. That ratio, 2(1 + √2) / (1 + √2)² = 0.828427, is fixed for any size, which makes it a fast way to estimate waste before you order sheet stock.
Two Widths, One Shape
Like the hexagon, an octagon has two different widths, and confusing them is the standard error. Across flats is 2 × apothem = s(1 + √2) ≈ 2.4142s. Across corners is 2 × circumradius = s√(4 + 2√2) ≈ 2.6131s. The two differ by about 8.2%, which is a smaller gap than the hexagon's 15.5% — an octagon is closer to circular — but it is still enough to matter for a fit.
For a gazebo, decking frame or table, across corners is the space the object actually needs; across flats is the width of the flat face against a wall. When a supplier quotes a single "size" without saying which, the safe assumption for clearance is across corners, since it is the larger of the two. If you assume flats when they meant corners, the piece will not fit its opening.
Angles and the Reason Octagons Do Not Tile
Every interior angle in a regular octagon is 135°, every exterior angle is 45°, and the interior angles sum to (8 − 2) × 180° = 1,080°. That 135° figure explains why octagons cannot tile a plane on their own: three of them at a point would need 405°, which is more than a full turn, while two leave a 90° gap. That gap is exactly a square, which is why octagon-and-square paving is such a common pattern — the two shapes fill the plane together precisely because 135 + 135 + 90 = 360.
The 45° exterior angle also makes the octagon unusually easy to build. Each cut is a 22.5° mitre, which is a standard setting on any mitre saw, and eight of them close the shape. That practicality, more than any aesthetic reason, is why octagonal frames, planters and gazebo bases are so common in timber construction.
Where Regular Octagons Are Specified
The best-known example is the STOP sign, which in the United States is reserved exclusively for that message under the Federal Highway Administration's Manual on Uniform Traffic Control Devices. The shape carries meaning on its own, so it is recognisable from the back or under snow when the lettering cannot be read — a deliberate piece of redundancy in the design.
Beyond signage, octagons appear in umbrella canopies, gazebo and pavilion roofs, drum shells, mirror and window frames, and in paving patterns alongside squares. In each case the shape is a compromise: it is far closer to a circle than a square is, so it uses space efficiently, while still being buildable from straight cuts and flat stock.
Estimating Material for an Octagonal Build
An octagonal deck, planter or gazebo base needs three numbers, and this calculator gives you all of them. The perimeter tells you the total length of edging, trim or fascia to buy — but buy it as eight separate pieces of the side length, not as one long run, because each piece needs mitres at both ends and the offcuts are not reusable. Eight pieces of 497 mm each is 3.98 m of finished trim, which in practice means ordering closer to 5 m of stock once mitre waste is allowed for.
The area tells you the decking, paving or sheet coverage. The two widths tell you the footprint: across corners is the space the finished object occupies, so that is the number to check against the site, while across flats is what sits against a wall or fence.
For a frame built from eight equal pieces, remember that the side length is measured on the long edge of each mitred piece, not the short edge. A 22.5° mitre on both ends of a board of thickness t shortens the inner edge by 2t × tan(22.5°) ≈ 0.828t relative to the outer edge. On a 45 mm thick timber that is about 37 mm of difference, which is enough to leave visible gaps at every joint if you cut all eight pieces to the same short-edge length.
Once you have the area, take it to a materials tool rather than ordering the bare number. Our concrete calculator adds a depth for a slab base, and the paint calculator converts a finished surface area into coverage per coat.
How the Octagon Compares
For a fixed perimeter of 80 units, an octagon encloses about 482.84 square units. A hexagon of the same perimeter encloses about 461.88, a square 400, and a circle about 509.30. The octagon captures roughly 94.8% of the circle's area with only eight straight cuts, which is why it is the usual stopping point when approximating a round shape in flat material. Going to twelve or sixteen sides adds fabrication work for a few more percent.
For other shapes we have the hexagon calculator, the pentagon calculator, the circle calculator for the round limiting case, and the rectangle area calculator for the square you cut from. Use the area converter to move a finished area between metric and imperial units.
Arb Digital designs, builds and publishes calculators that answer a real question in seconds and rank for the searches behind them. See the full set, or talk to us about tools for your own audience.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Treating across flats as the side length — that is a factor of 2.414 out and inflates the area by nearly six times.
- Confusing the two widths — across corners is about 8.2% larger than across flats, which is the difference between fitting an opening and not.
- Marking corner cuts at a quarter of the square — the correct fraction is 0.292893, not 0.25, and a 0.25 mark-out gives an irregular eight-sided shape with unequal edges.
- Setting the saw to 45° — for eight pieces meeting in a closed ring, each mitre is half the 45° exterior angle, so 22.5°.
- Applying this formula to an irregular octagon — all eight sides and all eight angles must be equal for the constant to hold.
Related Free Tools From Arb Digital
Handle any number of sides with the polygon calculator, measure a boundary with the perimeter calculator, convert linear sizes with the length converter, check angle units with the angle converter, and take a finished area to a materials order with the tile calculator. Our free online tools hub lists everything we publish.
Frequently Asked Questions
Area equals 2(1 + √2) multiplied by the side length squared, which is approximately 4.828427 × s². An octagon with 10 cm sides has an area of about 482.84 cm².
Measure in 0.292893 of the square's side length from each corner, along both edges, then cut off the four corner triangles. The remaining octagon is regular and its across-flats width equals the original square's side.
82.84%. The four corner triangles remove just over 17% of the original square, and that ratio is the same at any size.
Each interior angle is 135° and each exterior angle is 45°. The eight interior angles sum to 1,080°.
22.5°. Each joint between two pieces has to turn through the 45° exterior angle, and that turn is shared equally between the two mitred ends.
Because their 135° interior angles do not divide into 360°. Two octagons at a point leave a 90° gap, which is exactly a square — which is why octagon-and-square paving patterns work so well together.
Across flats is the width between two opposite flat edges, about 2.4142 side lengths. Across corners is the point-to-point width, about 2.6131 side lengths. Use across corners when checking clearance.
