Octagon Calculator
Enter any one measurement of a regular octagon and the calculator works out every other property: area, perimeter, all three diagonals, circumradius, apothem, and the number of distinct diagonals. You can also flip to the metric or imperial unit of your choice and turn on a flooring cost estimate to price a tile or hardwood job in one step.
Formula
Worked example
Side a = 12 in: area = 4.8284 x 144 = 695.3 in^2. Perimeter = 96 in. Short diagonal = 1.8478 x 12 = 22.17 in, medium = 2.4142 x 12 = 28.97 in, long = 2.6131 x 12 = 31.36 in. Circumradius = 15.68 in, apothem = 14.49 in. For flooring at 1 ft^2 tiles priced at $3.50 each with 10% waste: area ~4.83 ft^2, rounded up to 6 tiles = $21.00.
Solve from any known measurement
A regular octagon is fully determined by any one of its measurements because every length is a fixed multiple of the side. The "Solve from" menu lets you start with whatever you already know: the side, the area, the perimeter, any of the three diagonal lengths, the circumradius, or the apothem. The calculator inverts the formula to find the side length first, then derives everything else from it. If you measured a tile across the widest point, for example, enter that as the long diagonal and read off every other dimension instantly.
Three distinct diagonals and how to choose
Unlike a square, an octagon has three types of diagonals because any two vertices can be separated by one, two, or three edges. The short diagonal d1 = a x sqrt(2+sqrt(2)) (~1.848a) skips one corner and is sometimes called the "second-shortest" diagonal. The medium diagonal d2 = a(1+sqrt(2)) (~2.414a) skips two corners and is twice the apothem, so it equals the inner width of the octagon measured flat to flat. The long diagonal d3 = a x sqrt(4+2*sqrt(2)) (~2.613a) runs straight across opposite vertices and is the diameter of the circumscribed circle. When you measure a stop sign, the dimension you read across opposite flat sides is the medium diagonal, not the long one, because stop signs are measured flat-to-flat.
Circumradius, apothem, and drawing an octagon
The circumradius R is the distance from the centre to any vertex (R ~= 1.307a). Setting a compass to R and walking it around a circle eight times at 45-degree intervals places every vertex exactly. The apothem r is the centre-to-side distance (r ~= 1.207a) and is the radius of the inscribed circle that touches every side at its midpoint. The ratio r/R = cos(22.5 degrees) ~= 0.9239 is constant for all regular octagons. To check an octagon floor tile for squareness, verify that all eight sides are equal and that the two diagonals you can measure (medium and long) match the formulas.
Flooring and tiling cost estimate
Turn on the flooring toggle to price an octagonal room or table surface. The calculator divides the floor area by the tile or board size, rounds up to the nearest whole piece, and applies your waste allowance (10 to 15 percent is typical for straight cuts; add more for diagonal layouts). The result is the number of tiles to buy and the total material cost before delivery or labour. Because octagonal rooms sometimes use small square infill tiles between octagonal ones, this estimate covers the octagon itself; price the infill separately by its own area.
Regular octagon properties as multiples of the side a
| Property | Formula | Multiplier |
|---|---|---|
| Area (A) | 2(1+sqrt(2)) x a^2 | ~4.8284 x a^2 |
| Perimeter (P) | 8a | 8 x a |
| Short diagonal (d1) | a x sqrt(2+sqrt(2)) | ~1.8478 x a |
| Medium diagonal (d2) | a x (1+sqrt(2)) | ~2.4142 x a |
| Long diagonal (d3) | a x sqrt(4+2*sqrt(2)) | ~2.6131 x a |
| Circumradius (R) | d3 / 2 | ~1.3066 x a |
| Apothem (r) | a(1+sqrt(2)) / 2 | ~1.2071 x a |
| Interior angle | (n-2) x 180 / n | 135 deg |
| Exterior angle | 360 / n | 45 deg |
| Total diagonals | n(n-3) / 2 | 20 |
Multiply your side length by the constant in the right column to get each measure. For area, multiply a^2.
Frequently asked questions
Can I find the side length if I only know the area?
Yes. Set "Solve from" to Area, enter the area and its unit, and the calculator inverts the formula a = sqrt(A / 4.82843) to give you the side length plus every other property. The same applies if you know any diagonal, the perimeter, the circumradius, or the apothem.
What is the area formula for a regular octagon?
For a regular octagon with side length a, the area is 2(1+sqrt(2)) times a squared, which equals approximately 4.8284 times a squared. A side of 10 inches gives an area of about 482.84 square inches (3.35 square feet).
Why does an octagon have three different diagonal lengths?
Two vertices of an octagon can be separated by one, two, or three edges. Each separation distance is a different fixed multiple of the side: the short diagonal (~1.848a) skips one corner, the medium diagonal (~2.414a) skips two, and the long diagonal (~2.613a) connects opposite corners across the full width.
What is the difference between circumradius and apothem?
The circumradius R is the distance from the centre to any vertex (corner), while the apothem r is the distance from the centre to the midpoint of any side. The circumradius is always slightly larger: R ~= 1.3066a and r ~= 1.2071a. Their ratio is always cos(22.5 deg) ~= 0.9239.
How do I calculate how many tiles I need for an octagonal floor?
Divide the floor area by the area one tile covers, round up to the nearest whole number, and add 10 to 15 percent for cuts and waste. The flooring toggle on this calculator does exactly that: enter the tile size and price, set your waste allowance, and read off the tile count and total cost.
What is the interior angle of a regular octagon?
Each interior angle of a regular octagon is exactly 135 degrees. The general formula is (n-2) x 180 / n; for n = 8 this is 6 x 180 / 8 = 135 degrees. The exterior angle is 360 / 8 = 45 degrees, which is why two octagon segments fit neatly against a square corner.