Basic Geometrical Ideas - Polygons

Definition of a Polygon: A polygon is a simple closed curve made up entirely of line segments. To visualize this, imagine a shape like a star or a square where every boundary is a straight line and the shape is completely closed with no openings.

Sides and Vertices: The line segments that form a polygon are called its sides. The point where any two sides meet is called a vertex (plural: vertices). For example, a triangle has $3$ sides and $3$ vertices, which look like the three corners of the shape.

Adjacent Sides and Vertices: Two sides that share a common vertex are called adjacent sides. Similarly, the endpoints of the same side are called adjacent vertices. In a rectangle, the horizontal bottom side and the vertical side meeting at the bottom-right corner are adjacent sides.

Diagonals: A diagonal is a line segment connecting two non-adjacent vertices of a polygon. If you look at a quadrilateral (a four-sided shape), you can draw two slanted lines inside it connecting opposite corners; these are the diagonals. Note that a triangle has no diagonals.

Classification of Polygons: Polygons are named based on the number of sides they have. A $3$-sided polygon is a Triangle, a $4$-sided polygon is a Quadrilateral, a $5$-sided polygon is a Pentagon, and a $6$-sided polygon is a Hexagon.

Interior and Exterior: Like any closed curve, a polygon has an interior (the region inside the boundary), the boundary itself, and an exterior (the region outside). Any point located inside the shape is said to be in the interior of the polygon.

Regular and Irregular Polygons: A polygon is called 'Regular' if all its sides are of equal length and all its angles are equal (like a square or an equilateral triangle). If the sides or angles are not equal, it is called an 'Irregular' polygon.