Polygons

A polygon is a plane geometric figure formed by a finite sequence of straight line segments joined end to end to close a loop. The segments are called sides, the junctions are vertices, and the interior is the bounded region. By convention, most introductory descriptions consider simple polygons that do not intersect themselves; polygons that cross themselves are called complex or star polygons.

Polygons are classified by the number of sides, giving names such as triangle (3 sides), quadrilateral (4), pentagon (5), and in general n-gon for n sides. They are also classified by shape: convex polygons have all interior angles less than 180 degrees and any line segment joining two interior points lies inside the polygon; concave polygons have at least one interior angle greater than 180 degrees. A polygon is regular if all sides are equal and all interior angles are equal.

The interior angle sum of an n-gon is (n−2)×180 degrees, and the sum of exterior angles is 360 degrees. A polygon can have diagonals, which are segments joining nonadjacent vertices; the number of diagonals is n(n−3)/2. The area of a polygon can be computed by several methods, with the shoelace formula or triangulation being common for general polygons; the perimeter is the sum of all side lengths.

Some polygons have special properties: a regular polygon may have a circumradius (a circle through all vertices) and, in some cases, an inradius (a circle tangent to all sides). Not all polygons are cyclic or tangential. Regular polygons may tessellate the plane in certain arrangements; among regular polygons, only the equilateral triangle, the square, and the regular hexagon tile the plane by copies of themselves.