Angles and Polygons Lecture Notes

Definition of a Polygon

• A polygon is a shape with straight sides.
• Regular polygons have all sides and angles equal.

Types of Regular Polygons

• Triangle: 3 sides
  • Equilateral triangle is the regular polygon
• Quadrilateral: 4 sides
  • Square is the regular polygon
• Pentagon: 5 sides
  • Regular pentagon
• Hexagon: 6 sides
  • Regular hexagon
• Heptagon: 7 sides
  • Regular heptagon
• Octagon: 8 sides
  • Regular octagon
• Nonagon: 9 sides
  • Regular nonagon
• Decagon: 10 sides
  • Regular decagon

Sum of Interior Angles

• Formula: [(n - 2) \times 180]
  • n is the number of sides of the polygon.
• Examples:
  • Triangle: [3-2 \times 180 = 180] degrees
  • Quadrilateral: 360 degrees
  • Pentagon: 540 degrees
  • Hexagon: 720 degrees
  • Heptagon: 900 degrees
  • Octagon: 1080 degrees
  • Nonagon: 1260 degrees
  • Decagon: 1440 degrees

Applying the Formula

• For 11 sides: [(11-2) \times 180 = 1620] degrees
• For a polygon with 20 sides, calculate sum of angles:
  • [20-2 \times 180 = 3240] degrees
• For a polygon with 12 sides, each angle:
  • Total = 1800 degrees
  • Each angle = [1800 \div 12 = 150] degrees

Problem Solving

• Given the sum of interior angles (e.g., 3960 degrees), find the number of sides:
  • Solve [(n - 2) \times 180 = 3960]
  • [n = 24] sides after solving

Exterior Angles

• Sum of exterior angles of any polygon = 360 degrees
• Each interior angle + exterior angle = 180 degrees
• Example with a pentagon:
  • Interior = 108 degrees
  • Exterior = 72 degrees
  • Confirmed: [72 \times 5 = 360] degrees

Examples

• For a regular polygon with exterior angle of 20 degrees:
  • Sum of exterior angles = 360 degrees
  • Number of sides = [360 \div 20 = 18] sides

Key Concepts

• Interior angles add up using [(n-2)\times 180]
• Sum of exterior angles always 360 degrees
• Each interior and exterior angle pair sum to 180 degrees