heptagon

A heptagon is a polygon with seven sides and seven vertices. If all sides and all interior angles are equal, it is a regular heptagon; otherwise, it is irregular. The name comes from the Greek hepta "seven" and gonia "angle".

In Euclidean geometry, the sum of the interior angles of any seven-sided polygon is (7−2)×180° = 900°, so each angle in a regular heptagon measures 900°/7 ≈ 128.571°. The exterior angle is 360°/7 ≈ 51.429°. A regular heptagon can be inscribed in a circle; its vertices lie on a common circumcircle of radius R. If the side length is s, then s = 2R sin(π/7).

Area formulas for a regular heptagon include A = (7/4) s^2 cot(π/7) and A = (7/2) R^2 sin(2π/7). These relate the polygon’s size to its side length or circumradius.

Construction of a regular heptagon with straightedge and compass is not possible in the classical sense, because 7 is not a product of a power of two and distinct Fermat primes. It can be constructed with more advanced tools such as a marked ruler (neusis) or approached by numerical methods.

Symmetry and tiling: The regular heptagon has dihedral symmetry D7 (order 14). It does not tile the plane by itself in a regular tiling, though heptagons can appear in semi-regular tilings with other polygons.