nonagon

A nonagon is a polygon with nine sides and nine interior angles. The name comes from the Greek roots for nine (nonas) and angle (gonia). In polygonal terminology, an n-gon denotes a polygon with n sides; thus a nonagon is the nine-sided case. The sum of interior angles in any nonagon is (9−2)×180 = 1260 degrees, and the exterior angles sum to 360 degrees.

For a regular nonagon, all sides and angles are equal. Each interior angle measures 140 degrees, and each central angle is 40 degrees. If the side length is s, the perimeter is 9s. The area is A = (9/4) s^2 cot(π/9) ≈ 6.184 s^2. The number of diagonals is 27, given by n(n−3)/2. The circumradius R and inradius r satisfy R = s/(2 sin(π/9)) and r = s/(2 tan(π/9)). The figure has dihedral symmetry, denoted D9 (order 18).

Nonagons may be regular or irregular. A regular nonagon is not constructible with classical straightedge and compass, because 9 includes a repeated Fermat-prime factor; it can be constructed with alternate methods such as a marked ruler (neusis). A regular nonagon does not tile the plane by itself since 140° interior angles do not divide 360°, though nonagons can appear in tilings that combine different polygons.