Octagon Calculator
From a single side length, get the area, the perimeter, the apothem, and the circumradius — every measurement that describes a regular eight-sided polygon.
One input, four answers
Enter the side length and the calculator returns the area (2(1+√2)s²), the perimeter (8s), the apothem (((1+√2)/2)s), and the circumradius ((√(4+2√2)/2)s) at once.
Regular octagons only
These formulas assume all eight sides and all eight angles are equal. An irregular eight-sided shape needs a different method, and your answers come back in the same unit you typed in (squared for the area).
What is an octagon calculator?
One side in, the whole octagon out
An octagon calculator turns a single measurement — the length of one side — into every number that describes a regular octagon: the area it covers, the distance around it (perimeter), the apothem (centre to the middle of an edge), and the circumradius (centre to a corner). Because a regular octagon is perfectly symmetric, all eight sides are equal, so that one input fixes everything else. Octagons are familiar shapes: the stop sign, gazebos and pavilions, octagonal tables and mirrors, umbrella canopies, and plenty of tiling patterns. Whether you are building a gazebo floor, cutting a tabletop, or finishing a worksheet, the side length is all you need.
Enter one side length in any unit to get the area, perimeter, apothem, and circumradius instantly.
A handful of short formulas, all built from the side length s and the square root of 2 (≈ 1.414).
area = 2 × (1 + √2) × s²A regular octagon is a square with its four corners cut off at 45°, which is where the √2 comes from. The perimeter is simply 8 × s. The apothem — the distance from the centre to the middle of any side — is ((1 + √2) / 2) × s. The circumradius — the distance from the centre out to a corner — is (√(4 + 2√2) / 2) × s, and it is always a little longer than the apothem.
Suppose you have a regular octagon with a side of 6.
Perimeter
8 × 6 = 48 — the distance all the way around.
Apothem and circumradius
apothem = ((1 + √2) / 2) × 6 = 7.242641, circumradius = (√(4 + 2√2) / 2) × 6 = 7.839378.
Area
2 × (1 + √2) × 6² = 173.823376 square units — the surface inside.
The outputs answer different practical questions. The area (about 173.823376 square units for s = 6) is how much surface the octagon covers — the gazebo deck, the tabletop, the patch of floor you would tile. The perimeter (48 here) is the edging or trim that runs around it. The apothem (about 7.242641) is the centre-to-edge distance, the radius of the largest circle that fits inside, and exactly half the flat-to-flat width of the octagon — handy for fitting an octagon into a square opening. The circumradius (about 7.839378) is the centre-to-corner distance, the radius of the smallest circle that fits around it — what you set on a compass to mark out the eight corners. A useful fact: the circumradius is always larger than the apothem, because a corner is farther from the centre than the middle of a side, and a regular octagon's flat-to-flat width (twice the apothem) is about 2.414 times the side length.
The formulas are exact, but a couple of practical points are worth keeping in mind.
Regular octagons and consistent units
These formulas describe a perfect regular octagon — eight equal sides and eight equal 135° angles. An irregular octagon (sides or angles that differ) will not match these results; split it into triangles instead. The side length is also unit-agnostic, so the answers are only meaningful if you keep one unit throughout: a side in centimetres gives an area in square centimetres and a circumradius in centimetres, never a mix.