Exploring Angles in Polygons

Aug 22, 2024

Understanding Interior and Exterior Angles in Polygons

Sum of Interior Angles

  • Formula: (n - 2) * 180
    • n = number of sides (or angles) in the polygon
  • Explanation:
    • Example with a polygon having 5 sides (pentagon):
      • Can be divided into 3 triangles
      • Sum of angles in triangles = 3 * 180 = 540 degrees
    • Therefore, sum of interior angles for a polygon with n sides = (n - 2) * 180

Measure of One Interior Angle

  • Formula: (n - 2) * 180 / n
  • Application:
    • Used for regular polygons (equal angles and equal side lengths)
    • Gives the measure of one interior angle by dividing total interior angle sum by number of angles.

Measure of One Exterior Angle

  • Formula: 360 / n
  • Application:
    • Used for regular polygons to find one exterior angle.

Sum of Exterior Angles

  • Always add up to 360 degrees regardless of polygon type (regular or irregular).
    • Visualization: Formed angles on the outside of the polygon when extended add up to 360 degrees (like a circle).

Example Problems

Example 1: Finding an Interior Angle

  • Given a polygon with 5 sides:
    • Calculate sum of angles:
      • (5 - 2) * 180 = 540 degrees
    • Given angles: 120, 80, 100, 90
    • Equation: 120 + 80 + 100 + 90 + x = 540
    • Solution:
      • 390 + x = 540
      • x = 150 degrees

Example 2: Finding an Exterior Angle

  • Given angles: 90, 90, 120, x
    • Equation: 90 + 90 + 120 + x = 360
    • Solution:
      • 300 + x = 360
      • x = 60 degrees

Example 3: Regular Decagon

  • Definition: A decagon has 10 sides
  • Exterior angle calculation:
    • 360 / 10 = 36 degrees
  • Interior angle relationships:
    • Interior angle + Exterior angle = 180 degrees
    • If exterior angle is 36 degrees, then interior angle = 180 - 36 = 144 degrees.

Example 4: Type of Polygon from Interior Angles

  • Given that sum of interior angles = 2,340 degrees
    • Use the interior angle formula:
      • 2340 = (n - 2) * 180
    • Solve:
      • 2340 / 180 = 13
      • n - 2 = 13
      • n = 15
    • Conclusion: It is a 15-gon.

Conclusion

  • Importance of understanding interior and exterior angles in polygons for problem-solving.
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