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Interior Angles of Regular Polygons by Lew W. S.

Interior Angles of Regular Polygons by Lew W. S.
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2019-07-25 11:40
PolygonAngleSequenceGeometric ConstructionSlider Animation
① This GeoGebra applet visually demonstrates how the interior angles of regular polygons vary with the number of sides, from triangles to 20-gons, and shows the relationship between interior and exterior angles. ② Core math concepts include the polygon interior angle sum formula (n-2)×180°, each interior angle as (n-2)×180°/n, the constant exterior angle 360°/n, and the dynamic interplay of vertices, edges, and angles. ③ Construction highlights: using GeoGebra commands Intersect, Segment, Ray, Sequence, and Angle to create polygon vertices, edges, rays, and angle objects; a slider n controls the side count, generating lists VerticesPolygon, EdgesofPolygon, InternalAnglesPolygon, and ExternalAnglesPolygon; PolygonNamelist3to20 assigns names to polygons with 3 to 20 sides. ④ Suitable for middle school to high school geometry classroom demonstrations or student self-directed exploration, helping learners understand the derivation of the interior angle formula and compare regular with irregular polygons.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Segment(A, P) → a
  3. Segment((3.9999999999999933; 15°), (4; 30°)) → c
  4. Segment(C, D) → d
  5. Ray(C, B) → b
  6. Ray(B, C) → e
  7. Sequence(Segment[(r; i°), (r; (i - 360 / n)°)], i, 0, 360, 360 / n) → EdgesofPolygon
  8. Sequence(Ray[(r; i°), (r; (i - 360 / n)°)], i, 0, 360, 360 / n) → RaysofEdges
  9. Sequence(Angle[Element[RaysofEdges, i], Element[RaysofEdges, i + 1]], i, 1, n, 1) → ExternalAnglesPolygon
  10. Sequence(Angle[Element[RaysofEdges, i], Element[RaysofEdges, i + 1]], i, 1, n, 1) → ExternalAngles
  11. Angle(Element[RaysofEdges, 1], Element[RaysofEdges, 2]) → γ
  12. Angle(Element[RaysofEdges, 2], Element[RaysofEdges, 3]) → δ
  13. Angle(Element[RaysofEdges, 5], Element[RaysofEdges, 6]) → η
  14. Angle(Element[RaysofEdges, 3], Element[RaysofEdges, 4]) → ε
  15. Angle(Element[RaysofEdges, 4], Element[RaysofEdges, 5]) → ζ
  16. Angle(Element[RaysofEdges, 6], Element[RaysofEdges, 7]) → θ
  17. Angle(Element[RaysofEdges, 7], Element[RaysofEdges, 8]) → ι
  18. Angle(Element[RaysofEdges, 8], Element[RaysofEdges, 9]) → κ
  19. Angle(Element[RaysofEdges, 9], Element[RaysofEdges, 10]) → λ
  20. Angle(Element[RaysofEdges, 10], Element[RaysofEdges, 11]) → μ
  21. Angle(Element[RaysofEdges, 11], Element[RaysofEdges, 12]) → ν
  22. Angle(Element[RaysofEdges, 12], Element[RaysofEdges, 13]) → ξ
  23. Angle(Element[RaysofEdges, 13], Element[RaysofEdges, 14]) → ο
  24. Angle(Element[RaysofEdges, 14], Element[RaysofEdges, 15]) → ρ
  25. Angle(Element[RaysofEdges, 15], Element[RaysofEdges, 16]) → σ
  26. Angle(Element[RaysofEdges, 16], Element[RaysofEdges, 17]) → τ
  27. Angle(Element[RaysofEdges, 17], Element[RaysofEdges, 18]) → υ
  28. Angle(Element[RaysofEdges, 18], Element[RaysofEdges, 19]) → φ
  29. Angle(Element[RaysofEdges, 19], Element[RaysofEdges, 20]) → χ
  30. Angle(Element[RaysofEdges, 20], Element[RaysofEdges, 1]) → β
  31. Sequence((r; i°), i, 0, 360, 360 / n) → VerticesPolygon
  32. Sequence(Angle[(r; (i + (360 * 2 / n))°), (r; (i + 360 / n)°), (r; i°)], i, 0, 360, 360 / n) → InternalAnglesPolygon
  33. Angle(Element[VerticesPolygon, 3], Element[VerticesPolygon, 2], Element[VerticesPolygon, 1]) → α_1
  34. Angle(Element[VerticesPolygon, 4], Element[VerticesPolygon, 3], Element[VerticesPolygon, 2]) → β_2
  35. Angle(Element[VerticesPolygon, 5], Element[VerticesPolygon, 4], Element[VerticesPolygon, 3]) → ψ
  36. Angle(Element[VerticesPolygon, 6], Element[VerticesPolygon, 5], Element[VerticesPolygon, 4]) → ω
  37. Angle(Element[VerticesPolygon, 7], Element[VerticesPolygon, 6], Element[VerticesPolygon, 5]) → β_1
  38. Angle(Element[VerticesPolygon, 8], Element[VerticesPolygon, 7], Element[VerticesPolygon, 6]) → γ_1
  39. Angle(Element[VerticesPolygon, 9], Element[VerticesPolygon, 8], Element[VerticesPolygon, 7]) → δ_1
  40. Angle(Element[VerticesPolygon, 10], Element[VerticesPolygon, 9], Element[VerticesPolygon, 8]) → ε_1
  41. Angle(Element[VerticesPolygon, 11], Element[VerticesPolygon, 10], Element[VerticesPolygon, 9]) → ζ_4
  42. Angle(Element[VerticesPolygon, 13], Element[VerticesPolygon, 12], Element[VerticesPolygon, 11]) → η_1
  43. Angle(Element[VerticesPolygon, 12], Element[VerticesPolygon, 11], Element[VerticesPolygon, 10]) → θ_1
  44. Angle(Element[VerticesPolygon, 2], Element[VerticesPolygon, 1], Element[VerticesPolygon, n]) → ι_1
  45. Angle(Element[VerticesPolygon, 14], Element[VerticesPolygon, 13], Element[VerticesPolygon, 12]) → κ_1
  46. Angle(Element[VerticesPolygon, 15], Element[VerticesPolygon, 14], Element[VerticesPolygon, 13]) → λ_1
  47. Angle(Element[VerticesPolygon, 16], Element[VerticesPolygon, 15], Element[VerticesPolygon, 14]) → μ_1
  48. Angle(Element[VerticesPolygon, 17], Element[VerticesPolygon, 16], Element[VerticesPolygon, 15]) → ν_1
  49. Angle(Element[VerticesPolygon, 18], Element[VerticesPolygon, 17], Element[VerticesPolygon, 16]) → ξ_1
  50. Angle(Element[VerticesPolygon, 19], Element[VerticesPolygon, 18], Element[VerticesPolygon, 17]) → ο_1
  51. Angle(Element[VerticesPolygon, 20], Element[VerticesPolygon, 19], Element[VerticesPolygon, 18]) → ρ_1
  52. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 20], Element[VerticesPolygon, 19]) → σ_1
  53. Sequence(Segment[(r; 0°), (r; (i + 360 / n)°)], i, 360 / n, 360, 360 / n) → InsideTriangles
  54. Angle(Element[VerticesPolygon, 2], Element[VerticesPolygon, 1], Element[VerticesPolygon, 3]) → Δu_2
  55. Angle(Element[VerticesPolygon, 3], Element[VerticesPolygon, 2], Element[VerticesPolygon, 1]) → Δu
  56. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, n], Element[VerticesPolygon, n - 1]) → Δφ_1
  57. Angle(Element[VerticesPolygon, 3], Element[VerticesPolygon, 1], Element[VerticesPolygon, 4]) → Δχ_1
  58. Angle(Element[VerticesPolygon, 4], Element[VerticesPolygon, 1], Element[VerticesPolygon, 5]) → Δψ_1
  59. Angle(Element[VerticesPolygon, 5], Element[VerticesPolygon, 1], Element[VerticesPolygon, 6]) → Δω_1
  60. Angle(Element[VerticesPolygon, 6], Element[VerticesPolygon, 1], Element[VerticesPolygon, 7]) → Δα_2
  61. Angle(Element[VerticesPolygon, 7], Element[VerticesPolygon, 1], Element[VerticesPolygon, 8]) → Δγ_2
  62. Angle(Element[VerticesPolygon, 8], Element[VerticesPolygon, 1], Element[VerticesPolygon, 9]) → Δδ_2
  63. Angle(Element[VerticesPolygon, 9], Element[VerticesPolygon, 1], Element[VerticesPolygon, 10]) → Δε_2
  64. Angle(Element[VerticesPolygon, 10], Element[VerticesPolygon, 1], Element[VerticesPolygon, 11]) → Δζ_2
  65. Angle(Element[VerticesPolygon, 11], Element[VerticesPolygon, 1], Element[VerticesPolygon, 12]) → Δη_2
  66. Angle(Element[VerticesPolygon, 12], Element[VerticesPolygon, 1], Element[VerticesPolygon, 13]) → Δθ_2
  67. Angle(Element[VerticesPolygon, 13], Element[VerticesPolygon, 1], Element[VerticesPolygon, 14]) → Δι_2
  68. Angle(Element[VerticesPolygon, 14], Element[VerticesPolygon, 1], Element[VerticesPolygon, 15]) → Δκ_2
  69. Angle(Element[VerticesPolygon, 15], Element[VerticesPolygon, 1], Element[VerticesPolygon, 16]) → Δλ_2
  70. Angle(Element[VerticesPolygon, 16], Element[VerticesPolygon, 1], Element[VerticesPolygon, 17]) → Δμ_2
  71. Angle(Element[VerticesPolygon, 17], Element[VerticesPolygon, 1], Element[VerticesPolygon, 18]) → Δν_2
  72. Angle(Element[VerticesPolygon, 18], Element[VerticesPolygon, 1], Element[VerticesPolygon, 19]) → Δξ_2
  73. Angle(Element[VerticesPolygon, 19], Element[VerticesPolygon, 1], Element[VerticesPolygon, 20]) → Δο_2
  74. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 3], Element[VerticesPolygon, 2]) → Δu_3
  75. Angle(Element[VerticesPolygon, 4], Element[VerticesPolygon, 3], Element[VerticesPolygon, 1]) → Δσ_2
  76. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 4], Element[VerticesPolygon, 3]) → Δτ_2
  77. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 5], Element[VerticesPolygon, 4]) → Δυ_2
  78. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 6], Element[VerticesPolygon, 5]) → Δφ_2
  79. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 7], Element[VerticesPolygon, 6]) → Δχ_2
  80. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 8], Element[VerticesPolygon, 7]) → Δψ_2
  81. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 9], Element[VerticesPolygon, 8]) → Δω_2
  82. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 10], Element[VerticesPolygon, 9]) → Δα_3
  83. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 11], Element[VerticesPolygon, 10]) → Δβ_3
  84. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 12], Element[VerticesPolygon, 11]) → Δγ_3
  85. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 13], Element[VerticesPolygon, 12]) → Δδ_3
  86. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 14], Element[VerticesPolygon, 13]) → Δε_3
  87. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 16], Element[VerticesPolygon, 15]) → Δη_3
  88. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 17], Element[VerticesPolygon, 16]) → Δθ_3
  89. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 18], Element[VerticesPolygon, 17]) → Δι_3
  90. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 19], Element[VerticesPolygon, 18]) → Δκ_3
  91. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 20], Element[VerticesPolygon, 19]) → Δλ_3
  92. Angle(Element[VerticesPolygon, 5], Element[VerticesPolygon, 4], Element[VerticesPolygon, 1]) → Δμ_3
  93. Angle(Element[VerticesPolygon, 7], Element[VerticesPolygon, 6], Element[VerticesPolygon, 1]) → Δξ_3
  94. Angle(Element[VerticesPolygon, 8], Element[VerticesPolygon, 7], Element[VerticesPolygon, 1]) → Δο_3
  95. Angle(Element[VerticesPolygon, 9], Element[VerticesPolygon, 8], Element[VerticesPolygon, 1]) → Δρ_3
  96. Angle(Element[VerticesPolygon, 10], Element[VerticesPolygon, 9], Element[VerticesPolygon, 1]) → Δσ_3
  97. Angle(Element[VerticesPolygon, 11], Element[VerticesPolygon, 10], Element[VerticesPolygon, 1]) → Δτ_3
  98. Angle(Element[VerticesPolygon, 12], Element[VerticesPolygon, 11], Element[VerticesPolygon, 1]) → Δυ_3
  99. Angle(Element[VerticesPolygon, 13], Element[VerticesPolygon, 12], Element[VerticesPolygon, 1]) → Δφ_3
  100. Angle(Element[VerticesPolygon, 14], Element[VerticesPolygon, 13], Element[VerticesPolygon, 1]) → Δχ_3
  101. Angle(Element[VerticesPolygon, 15], Element[VerticesPolygon, 14], Element[VerticesPolygon, 1]) → Δψ_3
  102. Angle(Element[VerticesPolygon, 16], Element[VerticesPolygon, 15], Element[VerticesPolygon, 1]) → Δω_3
  103. Angle(Element[VerticesPolygon, 17], Element[VerticesPolygon, 16], Element[VerticesPolygon, 1]) → Δα_4
  104. Angle(Element[VerticesPolygon, 18], Element[VerticesPolygon, 17], Element[VerticesPolygon, 1]) → Δβ_4
  105. Angle(Element[VerticesPolygon, 19], Element[VerticesPolygon, 18], Element[VerticesPolygon, 1]) → Δγ_4
  106. Angle(Element[VerticesPolygon, 20], Element[VerticesPolygon, 19], Element[VerticesPolygon, 1]) → Δδ_4
  107. Angle(Element[VerticesPolygon, 7], Element[VerticesPolygon, 6], Element[VerticesPolygon, 1]) → Δζ_5
  108. Angle(Element[VerticesPolygon, 6], Element[VerticesPolygon, 5], Element[VerticesPolygon, 1]) → Δτ_1
  109. Angle(Element[VerticesPolygon, 1], Element[VerticesPolygon, 15], Element[VerticesPolygon, 14]) → Δζ_1

AI Drawing Prompt Reference

Draw a regular polygon from a triangle to a 20-gon, showing how the interior angles change with the number of sides. Start by drawing the intersection point A of the x-axis and y-axis, then draw a line segment a from A to point P. Next, draw a series of rays and line segments to form the sides and vertices of the polygon. Use a slider n to control the number of sides and generate lists of vertices, edges, internal angles, and external angles. Finally, display the degree of each internal and external angle and their relationships.
The prompt above was compiled from the original drawing commands and is for reference only.