# Polygons and Their Properties

Any closed plane figure with*n*sides is known as a polygon. If all the sides and the angles of this polygon are equivalent, the polygon is called a regular polygon. Polygons can be convex or concave. The word “polygon” derives from the Greek

*poly*, meaning “many,” and

*gonia*, meaning “angle.” The most familiar type of polygon is the regular polygon, which is a convex polygon with equal side lengths and angles.

The generalization of a polygon into three dimensions is called a

*polyhedron*, and into four dimensions is called a*polychoron*.A convex polygon is a simple polygon that has the following features:

- Every internal angle is at most 180 degrees.
- Every line segment between the two vertices of the polygon does not go outside the polygon (i.e., it remains inside or on the boundary of the polygon). In other words, all the diagonals of the polygon remain inside its boundary.
- Every triangle is strictly a convex polygon.

Convex Polygon

If a simple polygon is not convex, it is called concave. At least one internal angle of a concave polygon is larger than 180°.

Concave Polygon

As we can see in the above figure, one of the internal angles is more than 180°.

Here onwards, all the discussion about polygon refer to regular polygons only.

Polygons are named on the basis of the number of sides they have. A list of some of the polygons are given below:

Number of Sides |
Name of the polygon |

3 | Triangle |

4 | Quadrilateral |

5 | Pentagon |

6 | Hexagon |

7 | Heptagon |

8 | Octagon |

9 | Nonagon |

10 | Decagon |

# Area and Perimeter of a Regular Polygon

Given, A

_{1}A

_{2}A

_{3}A

_{4}…A

_{n}is a with regular polygon, ‘

*n*’ sides.

A

_{1 }A_{2 }= A_{2 }A_{3 }= A_{3}A_{4}= …. = A_{(n−1)}A_{n}= a units_{1}= OA

_{2}(Circumradius) = R

- Perimeter (P) =
*na* - Area
- Area
- Area
- Area
- r (in radius)

# Properties of a Polygon

- Interior Angle + Exterior Angle = 180°
- The number of diagonals in an
*n*-sided polygon =*n*(*n –*3)/2 - The sum total of all the exteriors angles of any polygon = 360°
- The measure of each exterior angle of a regular polygon
- The ratio of the sides of a polygon to the diagonals of a polygon is 2 : (
*n –*3) - The ratio of the interior angle of a regular polygon to its exterior angle is (
*n –*2) : 2 - The sum total of all the interior angles of any polygon = (2
*n*– 4) × 90°

# Some frequently used polygons

Apart from triangles and quadrilaterals, regular hexagon and regular octagon are also worth mentioning.Regular hexagon In the figure given below, ABCDEF is a regular hexagon with each side measuring a unit. Point O inside the hexagon is the centre of the hexagon.

Sum of the interior angles = 720°

Each interior angle = 120°

Each exterior angle = 60°

Area

*Regular octagon*In the figure given below, ABCD-EFGH is a regular octagon with each side measuring ‘a’ unit.

Sum of the interior angles = 1080°

Each interior angle = 135°

Each exterior angle = 45°

Area