Types and Properties of Polygons

Definition

• A polygon is a closed two-dimensional shape formed by a finite number of line segments (sides) connected end-to-end.

Types of Polygons

1. Based on Sides:

   • Triangle: 3 sides
   • Quadrilateral: 4 sides
   • Pentagon: 5 sides
   • Hexagon: 6 sides
   • Heptagon: 7 sides
   • Octagon: 8 sides
   • Nonagon: 9 sides
   • Decagon: 10 sides
   • n-gon: Any polygon with n sides

2. Based on Angles:

   • Convex Polygon: All interior angles are less than 180°.
   • Concave Polygon: At least one interior angle is greater than 180°.

3. Based on Regularity:

   • Regular Polygon: All sides and angles are equal (e.g., equilateral triangle, square).
   • Irregular Polygon: Sides and/or angles are not equal.

Properties

• Sum of Interior Angles: The sum can be calculated using the formula: (n - 2) × 180°, where n is the number of sides.
• Exterior Angles: The sum of the exterior angles of any polygon is always 360°.
• Diagonals: The number of diagonals in a polygon can be calculated using the formula: n(n - 3)/2, where n is the number of sides.

Notable Polygons

• Regular Polygons: Equilateral triangle, square, regular pentagon.
• Irregular Polygons: Shapes like parallelograms, trapezoids.

Applications

• Used in various fields such as architecture, computer graphics, and geometry.
• Understanding polygons is crucial in studying more complex shapes and structures.

Area Formulas

• Triangle: Area = 1/2 × base × height
• Quadrilateral: Area = base × height (for rectangles) or use specific formulas for other quadrilaterals.
• Regular Polygon: Area = (1/4) × n × s² × (cot(π/n)), where n = number of sides and s = side length.

Perimeter

• The perimeter of a polygon is the sum of the lengths of its sides.

Visualization

• Diagrams can help in understanding different types of polygons, their properties, and the relationships between sides and angles.

Definition

• A polygon is a closed 2D shape made up of a finite number of line segments, called sides, that are connected end-to-end.

Types of Polygons

• By Sides:

  • Triangle has 3 sides.
  • Quadrilateral has 4 sides.
  • Pentagon has 5 sides.
  • Hexagon has 6 sides.
  • Heptagon has 7 sides.
  • Octagon has 8 sides.
  • Nonagon has 9 sides.
  • Decagon has 10 sides.
  • n-gon can refer to any polygon with n sides.

• By Angles:

  • Convex polygons have all interior angles less than 180°.
  • Concave polygons have at least one interior angle greater than 180°.

• By Regularity:

  • Regular polygons have all sides and angles equal, such as an equilateral triangle or square.
  • Irregular polygons have unequal sides and/or angles.

Properties

• The sum of interior angles is calculated with (n - 2) × 180°, where n is the number of sides.
• The total of the exterior angles of any polygon always equals 360°.
• The number of diagonals in a polygon can be found using n(n - 3)/2.

Notable Polygons

• Regular examples include equilateral triangles, squares, and regular pentagons.
• Irregular examples include various shapes such as parallelograms and trapezoids.

Applications

• Polygons are crucial in fields like architecture, computer graphics, and geometry.
• Understanding polygons is necessary for studying complex shapes and structures.

Area Formulas

• Triangle area is calculated as 1/2 × base × height.
• For quadrilaterals, area can be found using base × height for rectangles, or other specific formulas for different types.
• Regular polygon area can be calculated with the formula: (1/4) × n × s² × (cot(π/n)), where n is the number of sides and s is the side length.

Perimeter

• Perimeter is the total distance around a polygon, found by summing the lengths of all sides.

Visualization

• Diagrams enhance comprehension of polygon types, their properties, and the relationships between sides and angles.

Test your knowledge on the various types of polygons, including their classifications based on sides, angles, and regularity. Understand the key properties and formulas associated with polygons to excel in geometric concepts.