Convex and concave polygons

3.1.1 Convex and concave polygons

Description

Quick Overview

This section introduces the concepts of convex and concave polygons, defining their characteristics and clarifying the distinctions between them.

Standard

In Section 3.1.1, we explore polygons categorized as either convex or concave. Convex polygons have diagonals that lie entirely within their interior, while concave polygons have at least one diagonal that extends outside of them. The importance of understanding these distinctions is emphasized in the context of further studies in polygons.

Detailed

Convex and Concave Polygons

Polygons are defined as simple closed curves made up of line segments, and they can be classified into two main categories: convex and concave.

Convex Polygons

  • Definition: A polygon is considered convex if all line segments connecting any two points within the polygon remain completely inside it. Consequently, none of the diagonals (lines connecting non-adjacent vertices) extend outside the polygon itself.
  • Characteristics:
  • All interior angles are less than 180°.
  • Any diagonal drawn lies entirely inside the polygon.

Concave Polygons

  • Definition: A polygon is convex if at least one diagonal lies outside the polygon. This means that when connecting some vertices, the resulting line segments may extend into the exterior space.
  • Characteristics:
  • At least one interior angle is greater than 180°.
  • Some diagonals will cross the boundary of the polygon.

Importance

This distinction between convex and concave polygons is crucial for understanding more complex geometric concepts and properties, particularly in later sections that deal with regular and irregular polygons.

Key Concepts

  • Polygon: A closed figure formed by connecting line segments.

  • Convex Polygon: No diagonals extend outside; all angles are less than 180°.

  • Concave Polygon: At least one diagonal extends outside; at least one angle is greater than 180°.

Memory Aids

🎵 Rhymes Time

  • In a convex shape, all points stay tight, / Concave shapes stretch, a broader sight.

📖 Fascinating Stories

  • Once in Geometry Land, there lived two families: the Convexes, always staying safe inside their homes, and the Concaves, who loved to extend their arms outside, reaching for the sky!

🧠 Other Memory Gems

  • C for Convex means 'C for Closed' – all angles are less than 180°, while C for Concave means 'C for Cut-out' – where at least one angle pushes out!

🎯 Super Acronyms

Remember C.C. for Concave - where a path is Cut out, versus V.V. for Convex - where it's all Within!

Examples

  • A square and a triangle are examples of convex polygons.

  • A star shape or an arrowhead are examples of concave polygons.

Glossary of Terms

  • Term: Polygon

    Definition:

    A simple closed curve composed of line segments.

  • Term: Convex Polygon

    Definition:

    A polygon where all diagonals lie entirely inside and all interior angles are less than 180°.

  • Term: Concave Polygon

    Definition:

    A polygon where at least one diagonal lies outside and at least one interior angle is greater than 180°.