Any polygon has at least three sides, with no theoretical upper limit to the number of sides it can have. For this reason, a polygon can be described as an nn 𝑛 -gon, where nn
𝑛
represents the number of sides. The name "polygon" comes from the Greek words "poly" (meaning "many") and "gonia" (meaning "angle").
What defines a polygon?
A polygon is a two-dimensional, closed shape made up entirely of straight-line segments. It is defined by its sides and vertices (corners), which are equal in number.
Key rules of a polygon:
- Must be a closed shape. All sides must connect at their endpoints, or vertices.
- Must have straight sides. A polygon cannot contain any curved lines.
- Must have at least three sides. Because a polygon must be a closed shape, a minimum of three connected line segments is required.
Polygons by number of sides
The number of sides a polygon has determines its specific name. Below are some common examples:
- **3 sides:**Triangle
- **4 sides:**Quadrilateral
- **5 sides:**Pentagon
- **6 sides:**Hexagon
- **7 sides:**Heptagon
- **8 sides:**Octagon
- **9 sides:**Nonagon
- **10 sides:**Decagon
- **11 sides:**Hendecagon (or Undecagon)
- **12 sides:**Dodecagon
The infinite-sided polygon (Apeirogon)
While a polygon must have a finite, or definite, number of sides to be constructed in the real world, the concept of a polygon with an infinite number of sides is a topic in theoretical geometry. This shape is called an apeirogon, from the Greek word apeiron, meaning "infinite".
Key properties of an apeirogon:
- Infinite sides. It has a countably infinite number of sides and vertices.
- Not a closed figure. Unlike a traditional polygon, an apeirogon is not a closed figure and does not have a finite area.
- Approaches a circle. For practical purposes, as the number of sides of a regular polygon increases toward infinity, the shape visually and mathematically approaches a circle. Many computer graphics applications and other calculations approximate a circle with a polygon that has a very high number of sides.
Classification of polygons by characteristics
Beyond the number of sides, polygons can be classified by their overall characteristics.
Regular vs. irregular:
- Regular polygon: All sides are equal in length, and all interior angles are equal in measure. A square and an equilateral triangle are examples of regular polygons.
- Irregular polygon: A polygon where the sides and angles are not all equal. A rectangle is a common example of an irregular polygon, as its sides are not all the same length.
Convex vs. concave:
- Convex polygon: A polygon in which all interior angles are less than 180° and point outwards. All regular polygons are convex.
- Concave polygon: A polygon with at least one interior angle greater than 180°. It has at least one vertex that points inward. A simple arrowhead shape is an example of a concave polygon.
Simple vs. complex:
- Simple polygon: A polygon whose sides do not intersect each other.
- Complex polygon: A polygon whose sides cross over each other. A five-pointed star (pentagram) is a type of complex polygon.