Polygon
GeometryA 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. It must have at least three sides. Triangles, squares, pentagons, and hexagons are all polygons. Circles and shapes with curved sides are NOT polygons.
Example
A stop sign is an octagon ($8$ sides). A yield sign is a triangle ($3$ sides). A soccer ball patch is a hexagon ($6$ sides). All of these are polygons because they are closed shapes with all straight sides.
Key Insight
The word polygon comes from Greek: "poly" means many and "gon" means angle. A polygon has exactly as many angles as it has sides. A $5$-sided polygon has $5$ angles; a $10$-sided polygon has $10$ angles.
Definition
A polygon is a closed plane figure formed by three or more line segments (sides) that meet only at their endpoints (vertices). The interior angle sum of an $n$-gon $= (n-2) \times 180^\circ$. Polygons are named by their number of sides: triangle ($3$), quadrilateral ($4$), pentagon ($5$), hexagon ($6$), heptagon ($7$), octagon ($8$), nonagon ($9$), decagon ($10$).
Example
A hexagon has interior angle sum $= (6-2) \times 180 = 720^\circ$. A regular hexagon has each interior angle $= 720/6 = 120^\circ$. A decagon ($10$ sides): sum $= (10-2) \times 180 = 1440^\circ$; each regular interior angle $= 144^\circ$.
Key Insight
The formula $(n-2) \times 180^\circ$ comes from dividing any polygon into $(n-2)$ triangles by drawing diagonals from one vertex. Each triangle contributes $180^\circ$, giving the total. This triangulation method is used in computer graphics and finite element analysis.
Definition
A simple polygon is a closed polygonal chain that does not self-intersect. Its interior angle sum is $(n-2)\pi$ radians (provable by triangulation). The exterior angle sum of any convex simple polygon is $2\pi$. By Pick's theorem, for a lattice polygon, Area $= I + B/2 - 1$, where $I$ is interior lattice points and $B$ is boundary lattice points.
Example
Pick's theorem example: lattice polygon with $I=6$ interior points and $B=8$ boundary points: Area $= 6 + 4 - 1 = 9$. For a unit square: $I=0$, $B=4$: Area $= 0 + 2 - 1 = 1$. Confirmed.
Key Insight
Triangulation of polygons is foundational in computational geometry: every simple polygon can be triangulated into exactly $n-2$ triangles. This result, combined with efficient triangulation algorithms ($O(n \log n)$), underpins 3D rendering, mesh generation, and geographic information systems.