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正多边形的内切圆半径与周长关系

正多边形的内切圆半径与周长关系
几何画板及绘图内容加载中，请稍等……
2013-07-02 21:53
多边形向量几何作图
所属专题 Regular Polygons
本范例演示了从3边到40边的正多边形内切圆半径与周长之间的关系。通过调整边数，观察内切圆半径如何影响多边形的周长。核心数学概念包括多边形、内切圆和周长。使用了旋转、条件语句、序列、距离、线段、多边形和向量等GeoGebra命令。适合初中及以上学生学习多边形性质。

GeoGebra 命令及参数

  1. Rotate(Pt1, -α, Pt2) → Pt3R
  2. Rotate(Pt2, -α, Pt3R) → Pt4R
  3. Rotate(Pt3R, -α, Pt4R) → Pt5R
  4. Rotate(Pt4R, -α, Pt5R) → Pt6R
  5. Rotate(Pt5R, -α, Pt6R) → Pt7R
  6. Rotate(Pt6R, -α, Pt7R) → Pt8R
  7. Rotate(Pt7R, -α, Pt8R) → Pt9R
  8. Rotate(Pt8R, -α, Pt9R) → Pt10R
  9. Rotate(Pt9R, -α, Pt10R) → Pt11R
  10. Rotate(Pt10R, -α, Pt11R) → Pt12R
  11. Rotate(Pt11R, -α, Pt12R) → Pt13R
  12. Rotate(Pt12R, -α, Pt13R) → Pt14R
  13. Rotate(Pt13R, -α, Pt14R) → Pt15R
  14. Rotate(Pt14R, -α, Pt15R) → Pt16R
  15. Rotate(Pt15R, -α, Pt16R) → Pt17R
  16. Rotate(Pt16R, -α, Pt17R) → Pt18R
  17. Rotate(Pt17R, -α, Pt18R) → Pt19R
  18. Rotate(Pt18R, -α, Pt19R) → Pt20R
  19. Rotate(Pt19R, -α, Pt20R) → Pt21R
  20. Rotate(Pt20R, -α, Pt21R) → Pt22R
  21. Rotate(Pt2, α, Pt1) → Pt3L
  22. Rotate(Pt1, α, Pt3L) → Pt4L
  23. Rotate(Pt3L, α, Pt4L) → Pt5L
  24. Rotate(Pt4L, α, Pt5L) → Pt6L
  25. Rotate(Pt5L, α, Pt6L) → Pt7L
  26. Rotate(Pt6L, α, Pt7L) → Pt8L
  27. Rotate(Pt7L, α, Pt8L) → Pt9L
  28. Rotate(Pt8L, α, Pt9L) → Pt10L
  29. Rotate(Pt9L, α, Pt10L) → Pt11L
  30. Rotate(Pt10L, α, Pt11L) → Pt12L
  31. Rotate(Pt11L, α, Pt12L) → Pt13L
  32. Rotate(Pt12L, α, Pt13L) → Pt14L
  33. Rotate(Pt13L, α, Pt14L) → Pt15L
  34. Rotate(Pt14L, α, Pt15L) → Pt16L
  35. Rotate(Pt15L, α, Pt16L) → Pt17L
  36. Rotate(Pt16L, α, Pt17L) → Pt18L
  37. Rotate(Pt17L, α, Pt18L) → Pt19L
  38. Rotate(Pt18L, α, Pt19L) → Pt20L
  39. Rotate(Pt19L, α, Pt20L) → Pt21L
  40. If(sides > 3, floor(sides / 2), 2) → sidesL
  41. If(sides > 3, ceil(sides / 2) + 1, 2) → sidesR
  42. Sequence(Segment[Element[pointsR, i], Element[pointsR, i + 1]], i, 0, sidesR) → segmentsR
  43. Sequence(Segment[Element[pointsL, i], Element[pointsL, i + 1]], i, 0, sidesL) → segmentsL
  44. Distance(Element[pointsL, sidesL + 1], Element[pointsR, sidesR + 1]) → openDistanceValue
  45. Rotate(Center + (0, -2), -α, Pt2) → Center2R
  46. Rotate(Center + (0, -2), α, Pt1) → Center2L
  47. Rotate(Center2R, (-open)°, Pt3R) → Center3R
  48. Rotate(Center3R, (-open)°, Pt4R) → Center4R
  49. Rotate(Center4R, (-open)°, Pt5R) → Center5R
  50. Rotate(Center5R, (-open)°, Pt6R) → Center6R
  51. Rotate(Center6R, (-open)°, Pt7R) → Center7R
  52. Rotate(Center7R, (-open)°, Pt8R) → Center8R
  53. Rotate(Center8R, (-open)°, Pt9R) → Center9R
  54. Rotate(Center9R, (-open)°, Pt10R) → Center10R
  55. Rotate(Center10R, (-open)°, Pt11R) → Center11R
  56. Rotate(Center11R, (-open)°, Pt12R) → Center12R
  57. Rotate(Center12R, (-open)°, Pt13R) → Center13R
  58. Rotate(Center13R, (-open)°, Pt14R) → Center14R
  59. Rotate(Center14R, (-open)°, Pt15R) → Center15R
  60. Rotate(Center15R, (-open)°, Pt16R) → Center16R
  61. Rotate(Center16R, (-open)°, Pt17R) → Center17R
  62. Rotate(Center17R, (-open)°, Pt18R) → Center18R
  63. Rotate(Center18R, (-open)°, Pt19R) → Center19R
  64. Rotate(Center19R, (-open)°, Pt20R) → Center20R
  65. Rotate(Center20R, (-open)°, Pt21R) → Center21R
  66. Sequence(Polygon[Element[pointsR, i], Element[pointsR, i + 1], Element[centersR, i]], i, 0, sidesR) → trianglesR
  67. Rotate(Center2L, open°, Pt3L) → Center3L
  68. Rotate(Center3L, open°, Pt4L) → Center4L
  69. Rotate(Center4L, open°, Pt5L) → Center5L
  70. Rotate(Center5L, open°, Pt6L) → Center6L
  71. Rotate(Center6L, open°, Pt7L) → Center7L
  72. Rotate(Center7L, open°, Pt8L) → Center8L
  73. Rotate(Center8L, open°, Pt9L) → Center9L
  74. Rotate(Center9L, open°, Pt10L) → Center10L
  75. Rotate(Center10L, open°, Pt11L) → Center11L
  76. Rotate(Center11L, open°, Pt12L) → Center12L
  77. Rotate(Center12L, open°, Pt13L) → Center13L
  78. Rotate(Center13L, open°, Pt14L) → Center14L
  79. Rotate(Center14L, open°, Pt15L) → Center15L
  80. Rotate(Center15L, open°, Pt16L) → Center16L
  81. Rotate(Center16L, open°, Pt17L) → Center17L
  82. Rotate(Center17L, open°, Pt18L) → Center18L
  83. Rotate(Center18L, open°, Pt19L) → Center19L
  84. Rotate(Center19L, open°, Pt20L) → Center20L
  85. Sequence(Polygon[Element[pointsL, i], Element[pointsL, i + 1], Element[centersL, i - 1]], i, 0, sidesL) → trianglesL
  86. Segment(Center + (0, 0), Center + (0, -1)) → apothemSegment
  87. Segment(Element[pointsR, sidesR + 1] + (0.05, 0), Element[pointsR, sidesR + 1] + (0.55, 0)) → altitudeHashB
  88. Segment(Element[pointsR, sidesR + 1] + (0.05, 1), Element[pointsR, sidesR + 1] + (0.55, 1)) → altitudeHashT
  89. Polygon(Pt1PolyOutline, Pt2PolyOutline, sides) → polygonOutline, a_1, h, i, j, k, l, m, n, p, q, r, pt1polyoutline, s, t, b_1, c_1, d_1, e_1, f_1, g_1, h_1, i_1, a, j_1, k_1, l_1, m_1, n_1, p_1, q_1, r_1, s_1, t_1, b, a_2, A, B, C, D, E, F, G, H, I, c, J, K, L, M, N, O, P, Q, R, S, d, T, U, V, W, Z, A_1, B_1, C_1, D_1, E_1, e, F_1, G_1, H_1, I_1, J_1, K_1, L_1, M_1, N_1, f, g
  90. Vector((x(Center), -0.25), (x(Element[pointsL, sidesL + 1]), -0.25)) → openDistanceVectorL
  91. Vector((x(Center), -0.25), (x(Element[pointsR, sidesR + 1]), -0.25)) → openDistanceVectorR
  92. Vector(Midpoint[Element[pointsR, sidesR + 1] + (0.3, 0), Element[pointsR, sidesR + 1] + (0.3, 1)], Element[pointsR, sidesR + 1] + (0.3, 0)) → altitudeVectorB
  93. Vector(Midpoint[Element[pointsR, sidesR + 1] + (0.3, 0), Element[pointsR, sidesR + 1] + (0.3, 1)], Element[pointsR, sidesR + 1] + (0.3, 1)) → altitudeVectorT
  94. Segment((x(Element[pointsL, sidesL + 1]), -0.1), (x(Element[pointsL, sidesL + 1]), -0.4)) → openDistanceEndLSolid
  95. Segment((x(Element[pointsR, sidesR + 1]), -0.1), (x(Element[pointsR, sidesR + 1]), -0.4)) → openDistanceEndRSolid

AI 绘图提示语参考

绘制一个正多边形，展示其边长和中心到边的距离（正多边形中心线段）之间的关系。通过改变边数，观察中心线段长度如何影响正多边形的周长。正多边形中心线段与边的关系可以通过旋转和分段来表现，先绘制正多边形，然后计算并显示中心线段长度和周长，最后通过旋转和分段来强调正多边形的对称性和中心线段的作用。
以上提示语为根据原始绘图命令整理而来，仅供参考。