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Interior Angles of Regular Polygons

Interior Angles of Regular Polygons
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2020-01-09 10:59
PolygonAngleSequenceGeometric ConstructionSlider Animation
This GeoGebra example dynamically demonstrates how the interior angles of regular polygons change as the number of sides increases. The core mathematical concepts include the sum of interior angles formula (n-2)×180° and each interior angle being (n-2)×180°/n, along with the relationship between interior and exterior angles (supplementary). The construction uses commands such as Intersect, Segment, Ray, Sequence, and Angle to build polygon vertices, edges, rays, and angle objects. A slider n controls the number of sides, generating a list of polygons from triangle to icosagon (PolygonNamelist3to20), and computes both internal angles (InternalAnglesPolygon) and external angles (ExternalAnglesPolygon) with distinct color coding. This is suitable for middle school geometry classes to visually explore the angle formula, and also useful for high school review or math lab activities, allowing students to drag the slider and observe angle changes to foster inductive reasoning.

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 with a variable number of sides controlled by a slider, ranging from triangle to icosagon. The polygon is centered at the origin with vertices evenly spaced on a circle. Extend rays from each vertex along the edges outward to form exterior angles. Mark each interior and exterior angle with arcs, using distinct color sets for interior vs exterior angles. Include horizontal and vertical axes, and a segment from the center to one vertex. As the slider changes, all angles and side count update together, visually demonstrating the interior angle sum formula.
The prompt above was compiled from the original drawing commands and is for reference only.