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Angle Properties of Polygons by user3597

Angle Properties of Polygons by user3597
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2013-09-26 20:12
PolygonAngleGeometric ConstructionSlider Animation
In collections S2 Angles in Triangles and Polygons
This example demonstrates the angle properties of polygons, including the sum of interior and exterior angles. Key math concepts involved are the polygon angle sum formula, exterior angle theorem, and angle calculations. Construction highlights the use of GeoGebra commands like Intersect, Segment, Ray, Sequence, and Angle, along with objects such as α, n, A, P, a, c, B, C, D, d, b, e, r, EdgesofPolygon, RaysofEdges, ExternalAnglesPolygon, ExternalAngles, γ, δ, η, ε, ζ, θ, ι, κ, λ, μ, ν, ξ, ο, ρ, σ, τ, υ, φ, χ, PolygonNamelist3to20, β, VerticesPolygon, and InternalAnglesPolygon. Suitable for high school students to learn and understand polygonal geometric properties, ideal for classroom instruction or self-directed exploration.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Segment(A, P) → a
  3. Segment((3.9999999999999933; 15.0*°), (4.0; 30.0*°)) → c
  4. Segment(C, D) → d
  5. Ray(C, B) → b
  6. Ray(B, C) → e
  7. Sequence(Segment[(r; i°), (r; (i - 360.0 / n)°)], i, 0.0, 360.0, 360.0 / n) → EdgesofPolygon
  8. Sequence(Ray[(r; i°), (r; (i - 360.0 / n)°)], i, 0.0, 360.0, 360.0 / n) → RaysofEdges
  9. Sequence(Angle[Element[RaysofEdges, i], Element[RaysofEdges, i + 1.0]], i, 1.0, n, 1.0) → ExternalAnglesPolygon
  10. Sequence(Angle[Element[RaysofEdges, i], Element[RaysofEdges, i + 1.0]], i, 1.0, n, 1.0) → ExternalAngles
  11. Angle(Element[RaysofEdges, 1.0], Element[RaysofEdges, 2.0]) → γ
  12. Angle(Element[RaysofEdges, 2.0], Element[RaysofEdges, 3.0]) → δ
  13. Angle(Element[RaysofEdges, 5.0], Element[RaysofEdges, 6.0]) → η
  14. Angle(Element[RaysofEdges, 3.0], Element[RaysofEdges, 4.0]) → ε
  15. Angle(Element[RaysofEdges, 4.0], Element[RaysofEdges, 5.0]) → ζ
  16. Angle(Element[RaysofEdges, 6.0], Element[RaysofEdges, 7.0]) → θ
  17. Angle(Element[RaysofEdges, 7.0], Element[RaysofEdges, 8.0]) → ι
  18. Angle(Element[RaysofEdges, 8.0], Element[RaysofEdges, 9.0]) → κ
  19. Angle(Element[RaysofEdges, 9.0], Element[RaysofEdges, 10.0]) → λ
  20. Angle(Element[RaysofEdges, 10.0], Element[RaysofEdges, 11.0]) → μ
  21. Angle(Element[RaysofEdges, 11.0], Element[RaysofEdges, 12.0]) → ν
  22. Angle(Element[RaysofEdges, 12.0], Element[RaysofEdges, 13.0]) → ξ
  23. Angle(Element[RaysofEdges, 13.0], Element[RaysofEdges, 14.0]) → ο
  24. Angle(Element[RaysofEdges, 14.0], Element[RaysofEdges, 15.0]) → ρ
  25. Angle(Element[RaysofEdges, 15.0], Element[RaysofEdges, 16.0]) → σ
  26. Angle(Element[RaysofEdges, 16.0], Element[RaysofEdges, 17.0]) → τ
  27. Angle(Element[RaysofEdges, 17.0], Element[RaysofEdges, 18.0]) → υ
  28. Angle(Element[RaysofEdges, 18.0], Element[RaysofEdges, 19.0]) → φ
  29. Angle(Element[RaysofEdges, 19.0], Element[RaysofEdges, 20.0]) → χ
  30. Angle(Element[RaysofEdges, 20.0], Element[RaysofEdges, 1.0]) → β
  31. Sequence((r; i°), i, 0.0, 360.0, 360.0 / n) → VerticesPolygon
  32. Sequence(Angle[(r; (i + 360.0 * 2.0 / n)°), (r; (i + 360.0 / n)°), (r; i°)], i, 0.0, 360.0, 360.0 / n) → InternalAnglesPolygon
  33. Angle(Element[VerticesPolygon, 3.0], Element[VerticesPolygon, 2.0], Element[VerticesPolygon, 1.0]) → α_1
  34. Angle(Element[VerticesPolygon, 4.0], Element[VerticesPolygon, 3.0], Element[VerticesPolygon, 2.0]) → β_2
  35. Angle(Element[VerticesPolygon, 5.0], Element[VerticesPolygon, 4.0], Element[VerticesPolygon, 3.0]) → ψ
  36. Angle(Element[VerticesPolygon, 6.0], Element[VerticesPolygon, 5.0], Element[VerticesPolygon, 4.0]) → ω
  37. Angle(Element[VerticesPolygon, 7.0], Element[VerticesPolygon, 6.0], Element[VerticesPolygon, 5.0]) → β_1
  38. Angle(Element[VerticesPolygon, 8.0], Element[VerticesPolygon, 7.0], Element[VerticesPolygon, 6.0]) → γ_1
  39. Angle(Element[VerticesPolygon, 9.0], Element[VerticesPolygon, 8.0], Element[VerticesPolygon, 7.0]) → δ_1
  40. Angle(Element[VerticesPolygon, 10.0], Element[VerticesPolygon, 9.0], Element[VerticesPolygon, 8.0]) → ε_1
  41. Angle(Element[VerticesPolygon, 11.0], Element[VerticesPolygon, 10.0], Element[VerticesPolygon, 9.0]) → ζ_4
  42. Angle(Element[VerticesPolygon, 13.0], Element[VerticesPolygon, 12.0], Element[VerticesPolygon, 11.0]) → η_1
  43. Angle(Element[VerticesPolygon, 12.0], Element[VerticesPolygon, 11.0], Element[VerticesPolygon, 10.0]) → θ_1
  44. Angle(Element[VerticesPolygon, 2.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, n]) → ι_1
  45. Angle(Element[VerticesPolygon, 14.0], Element[VerticesPolygon, 13.0], Element[VerticesPolygon, 12.0]) → κ_1
  46. Angle(Element[VerticesPolygon, 15.0], Element[VerticesPolygon, 14.0], Element[VerticesPolygon, 13.0]) → λ_1
  47. Angle(Element[VerticesPolygon, 16.0], Element[VerticesPolygon, 15.0], Element[VerticesPolygon, 14.0]) → μ_1
  48. Angle(Element[VerticesPolygon, 17.0], Element[VerticesPolygon, 16.0], Element[VerticesPolygon, 15.0]) → ν_1
  49. Angle(Element[VerticesPolygon, 18.0], Element[VerticesPolygon, 17.0], Element[VerticesPolygon, 16.0]) → ξ_1
  50. Angle(Element[VerticesPolygon, 19.0], Element[VerticesPolygon, 18.0], Element[VerticesPolygon, 17.0]) → ο_1
  51. Angle(Element[VerticesPolygon, 20.0], Element[VerticesPolygon, 19.0], Element[VerticesPolygon, 18.0]) → ρ_1
  52. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 20.0], Element[VerticesPolygon, 19.0]) → σ_1
  53. Sequence(Segment[(r; 0.0°), (r; (i + 360.0 / n)°)], i, 360.0 / n, 360.0, 360.0 / n) → InsideTriangles
  54. Angle(Element[VerticesPolygon, 2.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 3.0]) → Δu_2
  55. Angle(Element[VerticesPolygon, 3.0], Element[VerticesPolygon, 2.0], Element[VerticesPolygon, 1.0]) → Δu
  56. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, n], Element[VerticesPolygon, n - 1.0]) → Δφ_1
  57. Angle(Element[VerticesPolygon, 3.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 4.0]) → Δχ_1
  58. Angle(Element[VerticesPolygon, 4.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 5.0]) → Δψ_1
  59. Angle(Element[VerticesPolygon, 5.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 6.0]) → Δω_1
  60. Angle(Element[VerticesPolygon, 6.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 7.0]) → Δα_2
  61. Angle(Element[VerticesPolygon, 7.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 8.0]) → Δγ_2
  62. Angle(Element[VerticesPolygon, 8.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 9.0]) → Δδ_2
  63. Angle(Element[VerticesPolygon, 9.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 10.0]) → Δε_2
  64. Angle(Element[VerticesPolygon, 10.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 11.0]) → Δζ_2
  65. Angle(Element[VerticesPolygon, 11.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 12.0]) → Δη_2
  66. Angle(Element[VerticesPolygon, 12.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 13.0]) → Δθ_2
  67. Angle(Element[VerticesPolygon, 13.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 14.0]) → Δι_2
  68. Angle(Element[VerticesPolygon, 14.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 15.0]) → Δκ_2
  69. Angle(Element[VerticesPolygon, 15.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 16.0]) → Δλ_2
  70. Angle(Element[VerticesPolygon, 16.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 17.0]) → Δμ_2
  71. Angle(Element[VerticesPolygon, 17.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 18.0]) → Δν_2
  72. Angle(Element[VerticesPolygon, 18.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 19.0]) → Δξ_2
  73. Angle(Element[VerticesPolygon, 19.0], Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 20.0]) → Δο_2
  74. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 3.0], Element[VerticesPolygon, 2.0]) → Δu_3
  75. Angle(Element[VerticesPolygon, 4.0], Element[VerticesPolygon, 3.0], Element[VerticesPolygon, 1.0]) → Δσ_2
  76. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 4.0], Element[VerticesPolygon, 3.0]) → Δτ_2
  77. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 5.0], Element[VerticesPolygon, 4.0]) → Δυ_2
  78. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 6.0], Element[VerticesPolygon, 5.0]) → Δφ_2
  79. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 7.0], Element[VerticesPolygon, 6.0]) → Δχ_2
  80. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 8.0], Element[VerticesPolygon, 7.0]) → Δψ_2
  81. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 9.0], Element[VerticesPolygon, 8.0]) → Δω_2
  82. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 10.0], Element[VerticesPolygon, 9.0]) → Δα_3
  83. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 11.0], Element[VerticesPolygon, 10.0]) → Δβ_3
  84. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 12.0], Element[VerticesPolygon, 11.0]) → Δγ_3
  85. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 13.0], Element[VerticesPolygon, 12.0]) → Δδ_3
  86. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 14.0], Element[VerticesPolygon, 13.0]) → Δε_3
  87. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 16.0], Element[VerticesPolygon, 15.0]) → Δη_3
  88. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 17.0], Element[VerticesPolygon, 16.0]) → Δθ_3
  89. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 18.0], Element[VerticesPolygon, 17.0]) → Δι_3
  90. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 19.0], Element[VerticesPolygon, 18.0]) → Δκ_3
  91. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 20.0], Element[VerticesPolygon, 19.0]) → Δλ_3
  92. Angle(Element[VerticesPolygon, 5.0], Element[VerticesPolygon, 4.0], Element[VerticesPolygon, 1.0]) → Δμ_3
  93. Angle(Element[VerticesPolygon, 7.0], Element[VerticesPolygon, 6.0], Element[VerticesPolygon, 1.0]) → Δξ_3
  94. Angle(Element[VerticesPolygon, 8.0], Element[VerticesPolygon, 7.0], Element[VerticesPolygon, 1.0]) → Δο_3
  95. Angle(Element[VerticesPolygon, 9.0], Element[VerticesPolygon, 8.0], Element[VerticesPolygon, 1.0]) → Δρ_3
  96. Angle(Element[VerticesPolygon, 10.0], Element[VerticesPolygon, 9.0], Element[VerticesPolygon, 1.0]) → Δσ_3
  97. Angle(Element[VerticesPolygon, 11.0], Element[VerticesPolygon, 10.0], Element[VerticesPolygon, 1.0]) → Δτ_3
  98. Angle(Element[VerticesPolygon, 12.0], Element[VerticesPolygon, 11.0], Element[VerticesPolygon, 1.0]) → Δυ_3
  99. Angle(Element[VerticesPolygon, 13.0], Element[VerticesPolygon, 12.0], Element[VerticesPolygon, 1.0]) → Δφ_3
  100. Angle(Element[VerticesPolygon, 14.0], Element[VerticesPolygon, 13.0], Element[VerticesPolygon, 1.0]) → Δχ_3
  101. Angle(Element[VerticesPolygon, 15.0], Element[VerticesPolygon, 14.0], Element[VerticesPolygon, 1.0]) → Δψ_3
  102. Angle(Element[VerticesPolygon, 16.0], Element[VerticesPolygon, 15.0], Element[VerticesPolygon, 1.0]) → Δω_3
  103. Angle(Element[VerticesPolygon, 17.0], Element[VerticesPolygon, 16.0], Element[VerticesPolygon, 1.0]) → Δα_4
  104. Angle(Element[VerticesPolygon, 18.0], Element[VerticesPolygon, 17.0], Element[VerticesPolygon, 1.0]) → Δβ_4
  105. Angle(Element[VerticesPolygon, 19.0], Element[VerticesPolygon, 18.0], Element[VerticesPolygon, 1.0]) → Δγ_4
  106. Angle(Element[VerticesPolygon, 20.0], Element[VerticesPolygon, 19.0], Element[VerticesPolygon, 1.0]) → Δδ_4
  107. Angle(Element[VerticesPolygon, 7.0], Element[VerticesPolygon, 6.0], Element[VerticesPolygon, 1.0]) → Δζ_5
  108. Angle(Element[VerticesPolygon, 6.0], Element[VerticesPolygon, 5.0], Element[VerticesPolygon, 1.0]) → Δτ_1
  109. Angle(Element[VerticesPolygon, 1.0], Element[VerticesPolygon, 15.0], Element[VerticesPolygon, 14.0]) → Δζ_1

AI Drawing Prompt Reference

Draw a polygon to demonstrate the relationship between its interior and exterior angles. Start by plotting the x and y axes and finding their intersection to determine a vertex. Connect the vertices to form the polygon's sides. Then, draw rays from each vertex to create the exterior and interior angles. Use a slider to adjust the number of sides of the polygon, showing how the sum of interior angles and the properties of exterior angles change. Label key angles, such as the first exterior angle, the second exterior angle, etc., and display the formula for the sum of interior angles.
The prompt above was compiled from the original drawing commands and is for reference only.