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多边形内角与外角性质

多边形内角与外角性质
几何画板及绘图内容加载中，请稍等……
2013-09-26 20:12
多边形角几何作图滑块动画
所属专题 S2 Angles in Triangles and Polygons
本范例演示了多边形的内角与外角性质。涉及的核心数学概念包括多边形内角和公式、外角和定理以及角度计算。作图要点使用了 GeoGebra 的 Intersect（交点）、Segment（线段）、Ray（射线）、Sequence（序列）和 Angle（角度）等命令与对象，通过动态演示多边形的边、顶点和角度关系。适合高中阶段学生学习和理解多边形的几何性质，可用于课堂教学或自主探究。

GeoGebra 命令及参数

  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 绘图提示语参考

画一个多边形，展示其内角和外角的关系。首先绘制坐标轴并交点确定顶点，然后连接顶点形成多边形边。接着从每个顶点出发绘制射线，形成外角和内角。使用滑块调节多边形的边数，展示不同边数下内角和外角的计算和性质。标注关键角度，如第一个外角、第二个外角等，以及内角和的公式展示。
以上提示语为根据原始绘图命令整理而来，仅供参考。