A nonagon (sometimes spelled enneagon) is a polygon with nine straight sides and nine vertices. It belongs to the family of n‑gons and is one of the simplest polygons beyond the common pentagon and hexagon. Nonagons appear in pure geometry problems and as decorative or structural elements in design.

Basic geometric facts

For any nine‑sided polygon, the sum of the interior angles equals 1260 degrees, obtained from the general formula (n−2)·180°. In a regular nonagon—one whose sides and interior angles are all equal—each interior angle measures 140° and each exterior angle measures 40°. A nonagon can be convex or concave and may be simple (non‑self‑intersecting) or star‑shaped.

Properties of the regular nonagon

  • Interior angle (each): 140°.
  • Sum of interior angles: 1260°.
  • Number of diagonals: 27, found by n(n−3)/2.
  • Can be partitioned into 7 triangles by drawing nonoverlapping diagonals from one vertex (n−2 triangles).
  • Symmetry: cyclic symmetry C9 and dihedral symmetry D9 (order 18) for the regular case.

Constructibility and star forms

A regular nonagon is not constructible with ruler and compass alone because 9 = 3^2 does not meet the classical constructibility criteria based on Fermat primes. However, exact constructions using neusis methods or approximate compass‑and‑straightedge constructions exist. Several star polygons can be formed by connecting every k‑th vertex; notable examples are the {9/2} and {9/4} star polygons, often called nonagrams or enneagrams in recreational geometry and design.

Uses, examples, and notable facts

Because a regular nonagon's interior angle (140°) does not divide 360°, regular nonagons cannot tile the plane by themselves in a periodic edge‑to‑edge tiling. Nonagonal shapes are used in tiling with other polygons, in ornamentation, and in geometric explorations. The name combines Latin/Greek roots meaning "nine" and "angle," and both terms "nonagon" and "enneagon" are accepted in English.

For diagrams, constructions, and interactive examples of nine‑sided polygons, see additional resources on nonagons.