What is the formula for the area of a regular octagon?

Area = 2(1 + square-root-of-2) times side-squared, which simplifies to about 4.8284 times s-squared. For a side of 10 cm the area is roughly 482.84 cm squared. The formula comes from dividing the octagon into eight congruent isosceles triangles and summing their areas.

How do I find the side length of an octagon if I only know the area?

Rearrange the area formula: s = square-root-of (area divided by 2(1 + square-root-of-2)), which is approximately square-root-of (area divided by 4.8284). The calculator does this automatically when you select Area from the drop-down.

What is the difference between the inradius and the circumradius?

The inradius (r) is the distance from the centre to the midpoint of any side — it equals (1 + square-root-of-2) divided by 2 times s, roughly 1.2071 times s. The circumradius (R) is the distance from the centre to any vertex — it equals square-root-of(4 + 2*square-root-of-2) divided by 2 times s, roughly 1.3066 times s. The inradius is always shorter because an edge midpoint is closer to the centre than a corner.

What is the flat-to-flat width of a regular octagon?

The flat-to-flat width (also called the span or across-flats distance) is the gap between two opposite parallel sides. It equals (1 + square-root-of-2) times s, roughly 2.4142 times s. Numerically it is twice the inradius. This is the measurement you would use, for example, when fitting a stop-sign template or a tiled floor panel.

What is the long diagonal of a regular octagon?

The longest diagonal runs vertex to vertex through the centre and equals square-root-of(4 + 2*square-root-of-2) times s, roughly 2.6131 times s. It is twice the circumradius. There are also medium diagonals (skipping one vertex) and short diagonals (connecting vertices two steps apart), but this calculator focuses on the longest.

Does this tool work for non-regular octagons?

No — all formulas assume a regular octagon (all eight sides equal, all interior angles exactly 135 degrees). If your octagon is irregular you would need to break it into triangles and sum their areas manually.

Octagon Area Calculator

A regular octagon has eight equal sides and eight equal interior angles of 135 degrees each. It appears everywhere — stop signs, bathroom floor tiles, umbrellas, bolts, architectural windows — yet calculating its area or checking whether a piece will fit a given space trips people up regularly because the formula involves the square root of two in an unfamiliar way.

This calculator removes that friction. Tell it any one measurement you already know — the side length, the area, the perimeter, the inradius (centre to edge), the circumradius (centre to vertex), the long diagonal or the flat-to-flat width — and it instantly derives all the others. Select your preferred unit (mm, cm, m, in, ft or more), choose how many decimal places matter, then expand Show working to see every algebraic step with your numbers substituted in.

How it works

Every property of a regular octagon is a fixed multiple of the side length s. The calculator first inverts whichever formula matches your chosen input to recover s, then applies the standard constants forward.

The key formulas, where s is the side length:

Area = 2(1 + sqrt(2)) * s squared, approximately 4.8284 * s squared
Perimeter = 8 * s
Inradius r (centre to edge midpoint) = (1 + sqrt(2)) / 2 * s, approximately 1.2071 * s
Circumradius R (centre to vertex) = sqrt(4 + 2*sqrt(2)) / 2 * s, approximately 1.3066 * s
Long diagonal d (vertex to vertex through centre) = sqrt(4 + 2*sqrt(2)) * s = 2R, approximately 2.6131 * s
Width w (flat to flat across opposite sides) = (1 + sqrt(2)) * s = 2r, approximately 2.4142 * s

The SVG diagram labels s, r and R directly on the shape so you can see exactly which dimension each formula refers to before entering a number.

Worked example

Suppose a stop sign has a flat-to-flat width of 750 mm and you need the area to quote sheet metal:

Select “Width — flat to flat (w)” and enter 750, unit = mm.
Side: s = 750 / (1 + sqrt(2)) = 750 / 2.41421… = 310.66 mm
Area: 4.82843 * 310.66 squared = 4.82843 * 96,510 = 465,990 mm squared (about 0.466 m squared)
Perimeter: 8 * 310.66 = 2485.3 mm

Width w (mm)	Side s (mm)	Area (mm squared)	Perimeter (mm)
300	124.26	74,558	994.1
600	248.53	298,234	1988.2
750	310.66	465,990	2485.3
1000	414.21	828,427	3313.7

Formula note

The 2(1 + sqrt(2)) coefficient in the area formula comes from dividing the octagon into eight congruent isosceles triangles, each with a central angle of 45 degrees. Each triangle has area (1/2) * R squared * sin(45 degrees). Summing eight of them and expressing R in terms of s via R = sqrt(4 + 2*sqrt(2)) / 2 * s, and then simplifying, yields exactly 2(1 + sqrt(2)) * s squared. The constant 1 + sqrt(2) appears repeatedly because it is the cotangent of 22.5 degrees (half the interior vertex angle), which governs all flat-to-centre distances in the octagon.

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