nonagons

A nonagon is a polygon with nine sides and nine angles. The term nonagon is common in English, while enneagon is an alternative form derived from Greek roots ennea for "nine" and gonia for "angle."

In any nine‑sided polygon the sum of interior angles is (n − 2) × 180 degrees, giving 7 × 180 = 1260 degrees. A nonagon can be convex or concave and may be regular or irregular.

A regular nonagon has all sides equal and all interior angles equal. Each interior angle measures 140 degrees, and the central angle subtended by each side is 40 degrees. A regular nonagon can be inscribed in a circle, with its vertices on a common circle. The symmetry group of a regular nonagon is the dihedral group D9, consisting of 9 rotations and 9 reflections.

Diagonals and area: A nonagon has 27 diagonals. For a regular nonagon with side length s, the area is A = (9/4) s^2 cot(π/9) ≈ 6.183 s^2. If the circumradius is used instead, area can be expressed in terms of that radius as well.

Construction: The regular nonagon is not constructible with straightedge and compass, because 9 is not a product of a power of two and distinct Fermat primes. Nevertheless, approximate constructions exist. In practice, nonagons appear mainly in mathematical examples, education, and decorative designs rather than as common geometric tiles.

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