Octagon Calculator
Solve a regular octagon from any one measurement: side length, width across the flats, span across the corners, perimeter, or area. Works in any unit — enter inches, feet, or meters and results come back the same way.
Example: with What do you know? Side length (a) · Value 10 → Area: 482.843 square units.
- Side (a)10 units
- Perimeter (8a)80 units
- Across flats (1+√2)a24.142 units
Computed by the calculator below using its default values. Change any input to see your own numbers.
Regular octagon: A = 2(1+√2)a² ≈ 4.828a². A standard US stop sign is 30 in across the flats — sides of about 12.4 in.
A square with the corners cut off
The cleanest way to see a regular octagon is as a square with four identical corner triangles removed. Cutting each corner of a square of width (1+√2)a at 45° leaves eight equal sides of length a, and subtracting the four triangles from the square gives the area formula A = 2(1+√2)a² ≈ 4.828a². Carpenters build octagon planters and gazebo decks exactly this way — cut the corners off a square at 22.5° miters.
Octagons are measured three different ways in the wild, which is why this page accepts all of them. Stop signs are specified across the flats (30 in for a standard US sign under the MUTCD); gazebo kits often quote point-to-point span, which is √(4+2√2)·a ≈ 2.613a; and lumber lists work from the side length. Any one of the three pins down the other two.
How it’s calculated
A = 2(1+√2)a² with 2(1+√2) ≈ 4.828427; perimeter P = 8a; width across flats w = (1+√2)a ≈ 2.414214a; span across corners d = √(4+2√2)·a ≈ 2.613126a. Reverse solving: a = w/2.414214, a = d/2.613126, a = P/8, a = √(A/4.828427).
Regular octagons only — eight equal sides and 135° interior angles. A stretched or hand-drawn octagon needs to be decomposed into simpler shapes.
Regular octagon dimensions
| Side a | Area | Across flats | Across corners |
|---|---|---|---|
| 1 | 4.83 | 2.41 | 2.61 |
| 5 | 120.71 | 12.07 | 13.07 |
| 10 | 482.84 | 24.14 | 26.13 |
| Stop sign: 12.43 in | 745.6 in² ≈ 5.2 ft² | 30 in | 32.5 in |
Computed with A = 2(1+√2)a²; stop-sign width of 30 in across flats per the FHWA MUTCD standard for single-lane roads.
Common mistakes
- Entering a point-to-point span as the across-flats width — the corner span is about 8% longer (2.613a vs 2.414a), enough to ruin a cut list.
- Treating the octagon like a circle and using πr² on half the span; that overstates area by roughly 10%.
- Assuming the side equals the flat-to-flat width divided by 3 — the real divisor is 1+√2 ≈ 2.414.
- Doubling the side and expecting double the area; it quadruples, since the side is squared.
Frequently asked questions
What is the area formula for an octagon?
For a regular octagon with side a, A = 2(1+√2)a² ≈ 4.828a². It comes from a square of width (1+√2)a with the four corner triangles cut off.
How big is a stop sign?
A standard US stop sign is 30 inches across the flats (36 or 48 in on multi-lane and high-speed roads). That means sides of about 12.4 in, a point-to-point span of about 32.5 in, and roughly 5.2 square feet of face.
What is the difference between across flats and across corners?
Across flats runs from the middle of one side to the middle of the opposite side: (1+√2)a. Across corners runs point to opposite point: √(4+2√2)a, always about 8% longer. Spec sheets use both, so check which one you have.
How do I lay out an octagon from a square?
For a square of width w, mark w/(1+√2) ≈ 0.414w centered on each side and cut the corners at 45° between marks. Each cut removes a triangle whose legs are about 0.293w.
What are the interior angles of an octagon?
Each interior angle of a regular octagon is 135°, and the eight exterior angles are 45° each. Miter joints for an octagon frame are cut at half the exterior angle: 22.5°.