Polygons are closed chains of edges meeting at corners
A polygon is a plane figure whose line segments join into a closed polygonal chain. Sides are edges; meetings are vertices; an n-gon has n sides. Greek polus plus gonia means many-angled. Simple polygons refuse self-crossing; stars and skew space chains stretch the word.
A triangle is a 3-gon, and every polygon has as many corners as sides. The gon part may even trace back to the Greek word for knee. In a simple polygon the only meetings between segments are the shared endpoints of neighbours, and the region it encloses is called a solid polygon, whose interior is its body. Chains that cross themselves give star polygons, and some authors also count closed chains that leave a single plane, called skew polygons.
Convexity sorts them further. A line through a convex polygon, not merely grazing it, hits the boundary exactly twice, so every interior angle stays below 180 degrees, and this holds in any geometry. Concave polygons are simple but have at least one angle above 180 degrees. A star-shaped polygon has some point from which the whole interior is visible, which every convex polygon satisfies, yet no polygon can be both a star polygon and star-shaped. Rectilinear polygons meet only at 90 or 270 degrees.
Other labels describe symmetry and circles. Cyclic polygons have all corners on one circumcircle, and tangential ones have every side touching an inscribed circle. Vertex-transitive shapes are automatically cyclic and equiangular, edge-transitive ones equilateral and tangential, and a polygon is regular exactly when it is both, or equivalently when it is cyclic and equilateral; a non-convex regular one is a regular star polygon.
The interior angles of a simple n-gon add up to (n − 2) × 180 degrees, because it can be cut into n − 2 triangles. Walking once around a convex polygon turns you through one full revolution, so its exterior angles always total 360 degrees. Louis Poinsot first worked out the interior angles of regular star polygons, in the same paper that described the four regular star polyhedra.
Source: Polygon
