Classification of Polygons

Study Notes

Classification of Polygons

Definition of Polygons: A polygon is a closed, two-dimensional shape formed by a finite number of straight line segments (sides) that connect at points (vertices).
Regular Polygons:
- All sides are of equal length.
- All interior angles are equal.
- Examples: Equilateral triangle, square, regular pentagon.
Irregular Polygons:
- Sides are of different lengths.
- Interior angles are of different measures.
- Examples: Scalene triangle, rectangle (not a square), various other shapes.
Classification by Sides:
- Triangle: 3 sides
  - Regular: Equilateral triangle
  - Irregular: Scalene triangle
- Quadrilateral: 4 sides
  - Regular: Square
  - Irregular: Rectangle, trapezoid
- Pentagon: 5 sides
  - Regular: Regular pentagon
  - Irregular: Various shapes
- Hexagon: 6 sides
  - Regular: Regular hexagon
  - Irregular: Various shapes
- Continue classification for higher numbers of sides (heptagon, octagon, etc.).
Classification by Convexity:
- Convex Polygons: All interior angles are less than 180 degrees; no sides are indenting.
- Concave Polygons: At least one interior angle is greater than 180 degrees; at least one vertex points inward.
Classification by Symmetry:
- Symmetrical Polygons: Polygons that can be divided into two identical halves.
- Asymmetrical Polygons: Polygons that do not have identical halves when divided.
Complex Polygons:
- Self-Intersecting Polygons: Polygons that intersect themselves, such as star shapes (e.g., star polygon).
Useful Properties:
- The sum of interior angles of a polygon can be calculated using the formula:
  - Sum of interior angles = (n - 2) × 180°, where n is the number of sides.
- Regular polygons can be inscribed in and circumscribed around circles, demonstrating their symmetry and equal angles.

These classifications help in understanding the properties, calculations, and applications of different polygon types in geometry.

Definition and Basics

A polygon is a closed, two-dimensional shape created by connecting a finite number of straight line segments, known as sides, at points called vertices.

Types of Polygons

Regular Polygons:
- All sides and interior angles are equal.
- Examples include the equilateral triangle, square, and regular pentagon.
Irregular Polygons:
- Sides and interior angles can vary in length and measure.
- Examples consist of the scalene triangle and rectangle (not a square).

Classification by Number of Sides

Triangle:
- 3 sides
- Regular: Equilateral triangle
- Irregular: Scalene triangle
Quadrilateral:
- 4 sides
- Regular: Square
- Irregular: Rectangle, trapezoid
Pentagon:
- 5 sides
- Regular: Regular pentagon
- Irregular: Various other shapes
Hexagon:
- 6 sides
- Regular: Regular hexagon
- Irregular: Various other shapes
Higher classifications continue for heptagons, octagons, etc.

Classification by Convexity

Convex Polygons:
- All interior angles are less than 180 degrees and do not indent inward.
Concave Polygons:
- At least one interior angle exceeds 180 degrees, resulting in a vertex pointing inward.

Classification by Symmetry

Symmetrical Polygons:
- Can be divided into two identical halves.
Asymmetrical Polygons:
- Lack identical halves upon division.

Complex Polygons

Self-Intersecting Polygons:
- Intersect themselves, exemplified by star shapes.

Useful Properties

The formula for calculating the sum of interior angles is:
- Sum of interior angles = (n - 2) × 180°, where n represents the number of sides.
Regular polygons can be inscribed within and circumscribed around circles, illustrating their inherent symmetry and equal angles.

Application in Geometry

Understanding these classifications aids in comprehending the properties, calculations, and practical uses of various polygon types in geometry.

Description

This quiz explores the definitions and classifications of polygons, focusing on both regular and irregular shapes. You'll learn about various types of polygons categorized by their sides, including triangles, quadrilaterals, pentagons, and hexagons. Test your knowledge and solidify your understanding of these two-dimensional figures.