Look a little closer.

Eight sides, one consistent scale. These connections help explain why the figure behaves as it does, beyond simply entering numbers into a formula.

01 / A useful relationship

Cutting four equal right-isosceles corners from a suitable square can make a regular octagon.

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.

02 / A different perspective

A regular octagon’s area is 2(1+√2)a².

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.

03 / A connection to explore

Regular octagons can tile a plane together with squares.

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.

Try the idea with numbers

Open the from a side calculator, note its outputs, then multiply each length input by two. Area should increase by four and every reported length should double, when every relevant length is doubled. Angle inputs and dimensionless ratios remain unchanged. This scaling experiment checks the structure of the formula rather than the precision of any real measurement.

Further reading: Regular octagon at Wolfram MathWorld.