## What are Polygons?

Polygons are **2-dimensional** shapes.A polygon is a plane figure described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit in geometry**.** The term polygon is derived from the Greek words: poly (which means many) and gon (which means shape) (means angles). A polygonal circuit's segments are known as its **edges or sides**, and the points where two edges meet are known as the **polygon's vertices** (singular: vertex) or corners. As a result, an angle is formed. A solid polygon's interior is sometimes referred to as its **body**. An **n-gon** is a polygon with n sides, such as a triangle, which is a 3-gon. Polygons are classified primarily by the number of sides.

A polygon is a closed figure with all of its sides being line segments. Each side must intersect two other sides exactly once, but only at their endpoints. The sides must have a common endpoint and be noncollinear.

## Names of Polygon

Polygon | No. Of Sides | No. Of Diagonal | No. Of Vertices | Interior Angle Of Regular Polygon |

Triangle | 3 | 0 | 3 | 60 |

Quadrilateral | 4 | 2 | 4 | 90 |

Pentagon | 5 | 5 | 5 | 108 |

Hexagon | 6 | 9 | 6 | 120 |

Heptagon | 7 | 14 | 7 | 128.571 |

Octagon | 8 | 20 | 8 | 135 |

Nonagon | 9 | 27 | 9 | 140 |

Decagon | 10 | 35 | 10 | 144 |

Hendecagon | 11 | 44 | 11 | 147.273 |

Dodecagon | 12 | 54 | 12 | 150 |

Triskaidecagon | 13 | 65 | 13 | 152.308 |

Tetrakaidecagon | 14 | 77 | 14 | 154.286 |

Pentadecagon | 15 | 90 | 15 | 156 |

## Polygonal Shapes

Polygons are classified into several types based on their sides and angles, namely:

*Regular Polygon*

A regular polygon is one in which all of the sides and interior angles of the polygon are equal. Regular polygons include squares, equilateral triangles, and so on.

*Irregular Polygon*

An irregular polygon is one in which all of the sides and interior angles of the polygon are not of equal lengths. For instance, a scalene triangle, a rectangle, a kite, and so on.

*Convex Polygon*

A convex polygon is one in which all of the interior angles are strictly less than 180 degrees. The vertex will extend from the center of the shape.

*Concave polygon*

A concave polygon has one or more interior angles that are greater than 180 degrees. A concave polygon has a minimum of four sides. The vertex of the angle greater than 180 degrees is pointing inside the polygon.

*Simple Polygon*

A simple polygon has a single boundary that does not cross over itself.

*Complex Polygon*

A complex polygon has many boundaries that can cross over each other.

## Attributes of Polygon

*Perimeter of Polygon*

The sum of the lengths of all the sides of a polygon is called the perimeter of the polygon.

*Area of Polygon*

The area of a polygon is the region enclosed within a closed figure.

*Diagonal of Polygon*

A polygon's diagonal is the segment that connects any two non-consecutive vertices.

*Angles of Polygon*

Any polygon, as we know, has the same number of vertices as its sides. Each corner has a unique set of angles. These angles are divided into two types: interior and exterior angles of a polygon.

## Interior Angle Property

The sum of all the interior angles of a simple n-gon = (n âˆ’ 2) Ã— 180Â°

Or

Sum = (n âˆ’ 2)Ï€ radians

Where n denotes the number of sides of a polygon.

For example, because a quadrilateral has four sides, the sum of all interior angles is given by:

Sum of interior angles of 4-sided polygon = (4 â€“ 2) Ã— 180Â°

= 2 Ã— 180Â°

= 360Â°

## Exterior Angle Property

The interior and the corresponding exterior angles at each vertex of any polygon are supplementary to one another. In the case of a polygon;

Interior angle + Exterior angle = 180 degrees

Exterior angle = 180 degrees â€“ Interior angle

## Characteristics of Polygon

A polygon's properties are determined by its sides and angles.

- The sum of an n-sided polygon's interior angles is (n â€“ 2) 180Â°.
- The number of diagonals in an n-sided polygon = n(n â€“ 3)/2
- The number of triangles formed by connecting the diagonals from one of a polygon's corners = n â€“ 2.
- Each interior angle of an n-sided regular polygon is measured as [(n â€“ 2) 180Â°]/n.
- Each exterior angle of an n-sided regular polygon is measured as 360Â°/n.

## Triangles (3-gon)

A three-sided polygon is called a triangle. It is a fundamental shape in geometry. The sum of a triangle's three interior angles is always 180Â°. The sum of the lengths of a triangle's two sides is always greater than the length of the third side.

*Types of Triangles*

**Equilateral triangle-**A triangle whose all sides are of equal length and angles are of equal measure.**Isosceles triangle**- A triangle whose any 2 sides are equal and the angles opposite to the equal sides are equal.**Scalene triangle**- The triangle whose all three sides are unequal.

## Quadrilaterals (4-gon)

A quadrilateral is a polygon having the number of **sides equal to ****four***.* That means, a polygon formed by enclosing four line segments such that they meet at each other at corners/vertices to make 4 angles is a quadrilateral.

*Types of Quadrilateral*

**Square**-A square is a quadrilateral that has equal sides and each angle as a right angle (90Â°). In addition, opposite sides are parallel.**Rectangle**- A rectangle is a four-sided flat shape with all of its angles being right angles (90Â°). The opposite sides of a rectangle are parallel and equal in length.**Parallelogram**- A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel and equal in length. In addition, the opposite angles of a parallelogram are equal.**Rhombus**- A rhombus is a four-sided shape with equal length sides. In addition, opposite sides are parallel, and opposite angles are equal.**Trapezium**- A trapezoid is a four-sided flat shape with straight sides in which two opposite sides are parallel.

## Apothem of a Polygon

A regular polygon's apothem (sometimes abbreviated as apo) is a line segment that runs from the center to the midpoint of one of its sides. It is the line drawn from the polygon's center that is perpendicular to one of its sides. The term "apothem" can also refer to the length of the line segment in question. An apothem is shown in the following figure.

## Formulas

The area and perimeter formulas for various polygons are given below:

Name of Polygon | Area | Perimeter |
---|---|---|

Triangle | Â½ x (base) x (height) | a+b+c |

Square | side | 4 (side) |

Rectangle | Length x Breadth | 2(length+breadth) |

Parallelogram | Base x Height | 2(Sum of lengths of a pair of adjacent sides) |

Trapezoid | 1/2 (sum of parallel side)height | Sum of all sides |

Rhombus | Â½ (Product of diagonals) | 4 x side |

Regular Pentagon | $\frac{1}{2}\sqrt{5\left(5+2\sqrt{5}\right)}{\left(side\right)}^{2}$ | 5 x side |

Regular Hexagon | $.\frac{3\sqrt{3}}{2}{\left(side\right)}^{2}.$ | 6 x side |

## Common Mistakes

- Students might confuse between congruent and similar triangles.
- Errors while writing the ratio of corresponding sides.
- Errors while applying formulae to find the relation between the interior and the exterior angles of a regular polygon.

## Context & Applications

- A polygon is a primitive in computer graphics that is used in modeling and rendering.
- The tiles you walk on have a square shape, implying that they are polygons.
- Polygons include the truss of a building or bridge, the walls of a building, and so on.
- A polygon is the rectangular part of a chair in which you are sitting.

## Related Concepts

- Self-intersecting polygon
- Rectilinear polygon

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