How this calculator works
A regular octagon has eight equal sides and eight equal angles of 135 degrees. Its area can be found by adding a central square, four rectangles and four corner triangles, which simplifies to 2(1 + √2) times the square of the side. The apothem, the distance from the centre to a side, is (1 + √2)/2 times the side. Three diagonal lengths arise because a diagonal can skip one, two or three vertices. The width across flats is twice the apothem, while the longest diagonal and the width across corners are both twice the circumradius. Entering any one measurement first recovers the side and then builds every other value.
Formula
Side a. Perimeter = 8a. Area = 2(1 + √2)a². Apothem = a(1 + √2)/2. Circumradius = a/(2 sin 22.5°). Shortest diagonal = a√(2 + √2). Medium diagonal = a(1 + √2). Longest diagonal = a/sin 22.5°. Across flats = 2 × apothem.
Worked example
An octagon with side 10 m has perimeter 80 m, area 200(1 + √2) ≈ 482.8427 m², apothem 5(1 + √2) ≈ 12.0711 m, shortest diagonal 10√(2 + √2) ≈ 18.4776 m, medium diagonal 10(1 + √2) ≈ 24.1421 m and longest diagonal ≈ 26.1313 m.
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Frequently asked questions
What does across flats mean?
It is the distance between two opposite parallel sides, the same measurement used for the width of a nut or bolt head. It equals the medium diagonal.
Why does an octagon have three diagonal lengths?
A diagonal can skip one, two or three vertices before reaching the next one on a regular octagon. Skipping one gives the shortest diagonal, skipping two the medium diagonal, and skipping three passes through the centre and gives the longest diagonal.
How do I make an octagon from a square?
Cut equal corner triangles from a square. If the square has side S and the cut legs have length x, the remaining octagon has side x√2, so choosing x sets the final side.
Is the shortest diagonal the same as the across-flats width?
No. The shortest diagonal is a√(2 + √2), about 1.8478a, while the across-flats width is a(1 + √2), about 2.4142a.