Polygon
A polygon is a closed, flat figure made of three or more straight sides and angles, with the interior angle sum equal to (n-2) * 180 degrees.
Formula
\text{Interior angle sum} = (n-2) \times 180^\circ
Definition
A polygon is a flat, closed shape made entirely of straight sides, with at least three sides; triangles, squares, pentagons, and hexagons are all polygons, while circles and curved shapes are not. Formally, a simple polygon is a closed polygonal chain that does not self-intersect, formed by three or more line segments meeting only at their endpoints; the interior angle sum of an $n$-gon is $(n-2)\times180^\circ$ ($(n-2)\pi$ radians), provable by triangulation, and the exterior angle sum of any convex simple polygon is always $360^\circ$ ($2\pi$).
Example
A stop sign is an octagon ($8$ sides), a yield sign a triangle ($3$ sides), and a soccer-ball patch a hexagon ($6$ sides). A hexagon has interior angle sum $(6-2)\times180=720^\circ$, each regular interior angle $120^\circ$; a decagon's sum is $(10-2)\times180=1440^\circ$, each regular angle $144^\circ$. By Pick's theorem, for a lattice polygon, Area $=I+B/2-1$: with $6$ interior points and $8$ boundary points, area $=6+4-1=9$; a unit square ($I=0$, $B=4$) gives area $0+2-1=1$.
Key Insight
The word polygon comes from Greek, "poly" (many) and "gon" (angle), and a polygon has exactly as many angles as sides. The $(n-2)\times180^\circ$ formula comes from dividing any polygon into $(n-2)$ triangles by drawing diagonals from one vertex, a triangulation technique used in computer graphics, finite element analysis, 3D rendering, mesh generation, and geographic information systems, since every simple polygon can be triangulated this way in $O(n\log n)$ time.