Polygons are one of the most all-encompassing shapes in geometry. From the simple triangle up through squares, rectangles, trapezoids, to dodecagons and beyond.

In the figure above click on "make regular" and then gradually change the number of sides to see familiar shapes.

Types of Polygon

Regular		A polygon with all sides and interior angles the same. Regular polygons are always convex.  See Regular Polygons
Irregular		Each side may a different length, each angle may be a different measure. The opposite of a regular polygon. See Irregular Polygons
Convex		All interior angles less than 180°,and all vertices 'point outwards' away from the interior. The opposite of concave. Regular polygons are always convex. See Convex Polygons
Concave		One or more interior angles greater than 180°. Some vertices push 'inwards' towards the interior of the polygon. The opposite of convex. See Concave Polygons
Self-intersecting or Crossed		A polygon where one or more sides crosses back over another side, creating multiple smaller polygons. Most of the properties and theorems concerning polygons do not apply to this shape. It is best considered as several separate polygons. A polygon that in not self-intersecting in this way is called a simple polygon.

Properties of all Polygons (regular and irregular)

Interior angles	The interior angles of a polygon are those angles at each vertex on the inside of the polygon. See Polygon Interior Angles
Exterior Angles	The angle on the outside of a polygon between a side and the extended adjacent side.   See Polygon Exterior Angles
Diagonals	The diagonals of a polygon are lines linking any two non-adjacent vertices.   See Diagonals of a Polygon
Area	For regular polygons there are various ways to calculate the area. See Area of a Regular Polygon For irregular polygons things are a little harder since there no general formulae. For a general approach see Area of an Irregular Polygon
Perimeter	The distance around a polygon. The sum of its side lengths.   See Perimeter of a Polygon

Properties of Regular Polygons

Apothem (inradius)	The apothem of a polygon is a line from the center to the midpoint of a side. This is also the inradius - the radius of the incircle (see below).  See Apothem of a Polygon
Radius (circumradius)	The radius of a regular polygon is a line from the center to any vertex. It is also the radius of the circumcircle of the polygon (see below).  See Radius of a Polygon
Incircle	The incircle is the largest circle that will fit inside a regular polygon. Its radius is the apothem of the polygon (see above).   See Incircle of a Polygon
Circumcircle	The circle that passes through all the vertices of a regular polygon. Its radius is the radius of the polygon (see above).   See Circumcircle of a Polygon

Named Polygons

Many polygons have names based on the number of sides. A 5-sided polygon is called a pentagon for example. There are some that wish to name every possible polygon, but there seems little point in doing so. For example a 42-sided polygon is called a "tetracontakaidigon".

Beyond about 10 sides, most people call them an "n-gon". For example a 15-gon has 15 sides. This seems easier to remember and understand. However, there are some names that do occur in everyday experience:

Related polygon topics

General

• Polygon general definition
• Quadrilateral
• Regular polygon
• Irregular polygon
• Convex polygons
• Concave polygons
• Polygon diagonals
• Polygon triangles
• Apothem of a regular polygon
• Polygon center
• Radius of a regular polygon
• Incircle of a regular polygon
• Incenter of a regular polygon
• Circumcircle of a polygon
• Parallelogram inscribed in a quadrilateral

Types of polygon

• Square
• Diagonals of a square
• Rectangle
• Diagonals of a rectangle
• Golden rectangle
• Parallelogram
• Rhombus
• Trapezoid
• Trapezoid median
• Trapezium
• Kite

Area of various polygon types

• Regular polygon area
• Irregular polygon area
• Rhombus area
• Kite area
• Rectangle area
• Area of a square
• Trapezoid area
• Parallelogram area

Perimeter of various polygon types

• Perimeter of a polygon (regular and irregular)
• Perimeter of a triangle
• Perimeter of a rectangle
• Perimeter of a square
• Perimeter of a parallelogram
• Perimeter of a rhombus
• Perimeter of a trapezoid
• Perimeter of a kite

Angles associated with polygons

• Exterior angles of a polygon
• Interior angles of a polygon
• Relationship of interior/exterior angles
• Polygon central angle

Named polygons

• Tetragon, 4 sides
• Pentagon, 5 sides
• Hexagon, 6 sides
• Heptagon, 7 sides
• Octagon, 8 sides
• Nonagon Enneagon, 9 sides
• Decagon, 10 sides
• Undecagon, 11 sides
• Dodecagon, 12 sides