# polygon

Figure 1. Polygons are many-sided plane figures, the sides and angles of which may be equal or unequal, giving symmetrical or asymmetrical shapes. The simplest figure is the equilateral triangle (A1) which has all sides and angles equal. The isosceles triangle (A2) has two sides equal while (A3), which happens to be obtuse, has sides of differing length. The simplest quadrilateral (B1) is the square with equal sides and equal internal angles. The rectangle (B2) has two pairs of equal and opposite sides with equal internal angles. The parallelogram (B3) has two pairs of equal and opposite parallel sides. The trapezium (B4) has only two parallel sides. The regular pentagon (C) has five equal sides and the regular hexagon (D), six.

Figure 2. The polygon *ABCD* is an irregular quadrilateral
with acute angles at *A*, *C*, and *D* and a
reentrant angle at *B*. *AC* and *BD* are its
diagonals.

Figure 3. A regular polygon.

Figure 4. Regular polygons named.

A polygon is a plane closed figure whose sides are straight lines. The term *polygon* (from the Greek *poly* for "many" and *gwnos* for "angle") sometimes
also refers to the *interior* of the polygon (the open area that this
path encloses) or to the union of both.

A polygon is **simple** if it is described by a single, non-intersecting
boundary; otherwise it is said to be **complex**. A simple
polygon is called **convex** if it has no internal angles greater
than 180° otherwise it is called **concave**. A polygon
is called **regular** if all its sides are of equal length
and all its angles are equal. Any polygon, regular or irregular, has as
many angles as it has sides.

An *n*-gon is a polygon with *n* sides. For example, a hexagon is a 6-gon.

A **self-intersecting polygon** is a polygon with edges that cross other
edges.

Regular polygons | ||
---|---|---|

name | sides | interior angle = 180° - (360°/sides) |

equilateral triangle | 3 | 60° |

square | 4 | 90° |

regular pentagon | 5 | 108° |

regular hexagon | 6 | 120° |

regular heptagon | 7 | 128.57° (approx.) |

regular octagon | 8 | 135° |

regular nonagon | 9 | 140° |

regular decagon | 10 | 144° |

regular hectagon | 100 | 176.4° |

regular megagon | 10^{6} |
179.99964° |

regular googolgon | 10^{100} |
180° (approx.) |

## Formulae for regular polygons

In a regular *n*-sided polygon (see Figure 3), angle *A* = 180(1 - 2/*n*); angle *O* = 360/*n*; the area of the polygon = 1/2*nld*; and the radius *r* = *d*.sec(180/*n*).