# Polygons

A polygon is a two-dimensional geometric shape that is defined by a finite number of straight line segments, known as sides, that form a closed figure. The segments intersect only at their endpoints, called vertices. The term "polygon" comes from the Greek words "poly," meaning many, and "gonia," meaning angle. Polygons can be classified based on the number of sides they have.

Key Concepts:

Vertices:

The points where the sides of the polygon meet. Each vertex is a corner of the polygon.

Sides:

The straight line segments that form the edges or boundaries of the polygon. The sides do not intersect except at the vertices.

Interior and Exterior:

The interior of the polygon is the region enclosed by the sides, and the exterior is the region outside the polygon.

Diagonals:

Line segments connecting non-adjacent vertices of the polygon. The number of diagonals in a polygon can be calculated using the formula

�(�−3)/2

n(n−3)/2, where

�

n is the number of vertices.

Classification of Polygons:

Polygons can be classified based on the number of sides they have:

Triangle (3 sides):

The simplest polygon with three sides and three vertices.

Quadrilateral (4 sides):

A polygon with four sides and four vertices.

Pentagon (5 sides):

A polygon with five sides and five vertices.

Hexagon (6 sides):

A polygon with six sides and six vertices.

Heptagon (7 sides):

A polygon with seven sides and seven vertices.

Octagon (8 sides):

A polygon with eight sides and eight vertices.

Nonagon (9 sides):

A polygon with nine sides and nine vertices.

Decagon (10 sides):

A polygon with ten sides and ten vertices.

Regular and Irregular Polygons:

Regular Polygon:

A polygon with all sides of equal length and all angles of equal measure.

Irregular Polygon:

A polygon that does not satisfy the conditions of a regular polygon. It may have sides and angles of different lengths and measures.

Convex and Concave Polygons:

Convex Polygon:

All interior angles of the polygon are less than 180 degrees.

Concave Polygon:

At least one interior angle of the polygon is greater than 180 degrees.

Understanding polygons is fundamental in geometry, and their properties are used in various mathematical and architectural applications. The study of polygons involves analyzing their angles, sides, and relationships between vertices, contributing to a broader understanding of geometric shapes.