Understand the regular octagon

A regular octagon has eight equal sides and eight equal interior angles. A stop-sign outline is a familiar visual example, though this site calculates mathematical dimensions rather than any jurisdiction’s sign standard.

Properties that make the calculations work

Each interior angle is 135 degrees and each central angle is 45 degrees. Its apothem reaches a side midpoint, and its circumradius reaches a vertex. The perimeter is eight times a side.

Where the model is useful

Use the area inverse tool when the enclosed region is specified, or use the apothem when the drawing is dimensioned from its centre to a flat side. All tools use exact trigonometric relations before numerical rounding.

Measure the right thing

A general eight-sided polygon is not determined by one side. These tools apply only to regular octagons.

Sketch the figure and mark every known measurement before choosing a calculator. A length measured along an edge is not automatically a perpendicular height. If a formula needs a radius, check that the value has not been measured across a full diameter. Write the unit beside each known quantity before entering it.

Scale and units

Multiplying every length by k multiplies any area by k². This is a useful check: doubling all dimensions cannot merely double an area. Inputs expressed in centimetres produce cm² for area; the unit selector relabels rather than converts entered numbers.

Continue learning

Wolfram MathWorld: Regular octagon provides reference reading. Our summaries and examples are written for this portal and are not a specialist-reviewed textbook. Try a worked example or suggest a correction.