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格点多边形

格点多边形
几何画板及绘图内容加载中，请稍等……
2015-01-17 00:17
多边形解析几何
本范例通过探索格点多边形的面积与内部点数之间的关系，帮助学生理解多边形面积计算方法。涉及的核心数学概念包括多边形面积、点数统计和几何作图。使用了Polygon, If, CountIf, Point, Intersect等GeoGebra命令。适合初中及以上学生学习多边形面积计算和几何作图。

GeoGebra 命令及参数

  1. Polygon(P_1, P_2, P_3, P_4, P_5) → poly1, b_1, c_1, d_1, e_1, a_1
  2. If(IsInRegion[B, poly1], 1, 0) → b1
  3. If(IsInRegion[C, poly1], 1, 0) → c1
  4. If(IsInRegion[D, poly1], 1, 0) → d1
  5. If(IsInRegion[E, poly1], 1, 0) → e1
  6. If(IsInRegion[F, poly1], 1, 0) → f1
  7. If(IsInRegion[G, poly1], 1, 0) → g1
  8. If(IsInRegion[H, poly1], 1, 0) → h1
  9. If(IsInRegion[I, poly1], 1, 0) → i1
  10. If(IsInRegion[J, poly1], 1, 0) → j1
  11. If(IsInRegion[K, poly1], 1, 0) → k1
  12. If(IsInRegion[L, poly1], 1, 0) → l1
  13. If(IsInRegion[M, poly1], 1, 0) → m1
  14. If(IsInRegion[N, poly1], 1, 0) → n1
  15. If(IsInRegion[O, poly1], 1, 0) → o1
  16. If(IsInRegion[P, poly1], 1, 0) → p1
  17. If(IsInRegion[Q, poly1], 1, 0) → q1
  18. If(IsInRegion[R, poly1], 1, 0) → r1
  19. If(IsInRegion[S, poly1], 1, 0) → s1
  20. If(IsInRegion[T, poly1], 1, 0) → t1
  21. If(IsInRegion[U, poly1], 1, 0) → u1
  22. If(IsInRegion[V, poly1], 1, 0) → v1
  23. If(IsInRegion[W, poly1], 1, 0) → w1
  24. If(IsInRegion[Z, poly1], 1, 0) → z1
  25. If(IsInRegion[A, poly1], 1, 0) → a1
  26. If(IsInRegion[AA, poly1], 1, 0) → aa1
  27. If(Distance[A, a_1] ≟ 0, 1, If[Distance[A, b_1] ≟ 0, 1, If[Distance[A, c_1] ≟ 0, 1, If[Distance[A, d_1] ≟ 0, 1, If[Distance[A, e_1] ≟ 0, 1, 0]]]]) → a2
  28. If(Distance[B, a_1] ≟ 0, 1, If[Distance[B, b_1] ≟ 0, 1, If[Distance[B, c_1] ≟ 0, 1, If[Distance[B, d_1] ≟ 0, 1, If[Distance[B, e_1] ≟ 0, 1, 0]]]]) → b2
  29. If(Distance[C, a_1] ≟ 0, 1, If[Distance[C, b_1] ≟ 0, 1, If[Distance[C, c_1] ≟ 0, 1, If[Distance[C, d_1] ≟ 0, 1, If[Distance[C, e_1] ≟ 0, 1, 0]]]]) → c2
  30. If(Distance[D, a_1] ≟ 0, 1, If[Distance[D, b_1] ≟ 0, 1, If[Distance[D, c_1] ≟ 0, 1, If[Distance[D, d_1] ≟ 0, 1, If[Distance[D, e_1] ≟ 0, 1, 0]]]]) → d2
  31. If(Distance[E, a_1] ≟ 0, 1, If[Distance[E, b_1] ≟ 0, 1, If[Distance[E, c_1] ≟ 0, 1, If[Distance[E, d_1] ≟ 0, 1, If[Distance[E, e_1] ≟ 0, 1, 0]]]]) → e2
  32. If(Distance[F, a_1] ≟ 0, 1, If[Distance[F, b_1] ≟ 0, 1, If[Distance[F, c_1] ≟ 0, 1, If[Distance[F, d_1] ≟ 0, 1, If[Distance[F, e_1] ≟ 0, 1, 0]]]]) → f2
  33. If(Distance[G, a_1] ≟ 0, 1, If[Distance[G, b_1] ≟ 0, 1, If[Distance[G, c_1] ≟ 0, 1, If[Distance[G, d_1] ≟ 0, 1, If[Distance[G, e_1] ≟ 0, 1, 0]]]]) → g2
  34. If(Distance[H, a_1] ≟ 0, 1, If[Distance[H, b_1] ≟ 0, 1, If[Distance[H, c_1] ≟ 0, 1, If[Distance[H, d_1] ≟ 0, 1, If[Distance[H, e_1] ≟ 0, 1, 0]]]]) → h2
  35. If(Distance[I, a_1] ≟ 0, 1, If[Distance[I, b_1] ≟ 0, 1, If[Distance[I, c_1] ≟ 0, 1, If[Distance[I, d_1] ≟ 0, 1, If[Distance[I, e_1] ≟ 0, 1, 0]]]]) → i2
  36. If(Distance[J, a_1] ≟ 0, 1, If[Distance[J, b_1] ≟ 0, 1, If[Distance[J, c_1] ≟ 0, 1, If[Distance[J, d_1] ≟ 0, 1, If[Distance[J, e_1] ≟ 0, 1, 0]]]]) → j2
  37. If(Distance[K, a_1] ≟ 0, 1, If[Distance[K, b_1] ≟ 0, 1, If[Distance[K, c_1] ≟ 0, 1, If[Distance[K, d_1] ≟ 0, 1, If[Distance[K, e_1] ≟ 0, 1, 0]]]]) → k2
  38. If(Distance[L, a_1] ≟ 0, 1, If[Distance[L, b_1] ≟ 0, 1, If[Distance[L, c_1] ≟ 0, 1, If[Distance[L, d_1] ≟ 0, 1, If[Distance[L, e_1] ≟ 0, 1, 0]]]]) → l2
  39. If(Distance[M, a_1] ≟ 0, 1, If[Distance[M, b_1] ≟ 0, 1, If[Distance[M, c_1] ≟ 0, 1, If[Distance[M, d_1] ≟ 0, 1, If[Distance[M, e_1] ≟ 0, 1, 0]]]]) → m2
  40. If(Distance[N, a_1] ≟ 0, 1, If[Distance[N, b_1] ≟ 0, 1, If[Distance[N, c_1] ≟ 0, 1, If[Distance[N, d_1] ≟ 0, 1, If[Distance[N, e_1] ≟ 0, 1, 0]]]]) → n2
  41. If(Distance[O, a_1] ≟ 0, 1, If[Distance[O, b_1] ≟ 0, 1, If[Distance[O, c_1] ≟ 0, 1, If[Distance[O, d_1] ≟ 0, 1, If[Distance[O, e_1] ≟ 0, 1, 0]]]]) → o2
  42. If(Distance[P, a_1] ≟ 0, 1, If[Distance[P, b_1] ≟ 0, 1, If[Distance[P, c_1] ≟ 0, 1, If[Distance[P, d_1] ≟ 0, 1, If[Distance[P, e_1] ≟ 0, 1, 0]]]]) → p2
  43. If(Distance[Q, a_1] ≟ 0, 1, If[Distance[Q, b_1] ≟ 0, 1, If[Distance[Q, c_1] ≟ 0, 1, If[Distance[Q, d_1] ≟ 0, 1, If[Distance[Q, e_1] ≟ 0, 1, 0]]]]) → q2
  44. If(Distance[R, a_1] ≟ 0, 1, If[Distance[R, b_1] ≟ 0, 1, If[Distance[R, c_1] ≟ 0, 1, If[Distance[R, d_1] ≟ 0, 1, If[Distance[R, e_1] ≟ 0, 1, 0]]]]) → r2
  45. If(Distance[S, a_1] ≟ 0, 1, If[Distance[S, b_1] ≟ 0, 1, If[Distance[S, c_1] ≟ 0, 1, If[Distance[S, d_1] ≟ 0, 1, If[Distance[S, e_1] ≟ 0, 1, 0]]]]) → s2
  46. If(Distance[T, a_1] ≟ 0, 1, If[Distance[T, b_1] ≟ 0, 1, If[Distance[T, c_1] ≟ 0, 1, If[Distance[T, d_1] ≟ 0, 1, If[Distance[T, e_1] ≟ 0, 1, 0]]]]) → t2
  47. If(Distance[U, a_1] ≟ 0, 1, If[Distance[U, b_1] ≟ 0, 1, If[Distance[U, c_1] ≟ 0, 1, If[Distance[U, d_1] ≟ 0, 1, If[Distance[U, e_1] ≟ 0, 1, 0]]]]) → u2
  48. If(Distance[V, a_1] ≟ 0, 1, If[Distance[V, b_1] ≟ 0, 1, If[Distance[V, c_1] ≟ 0, 1, If[Distance[V, d_1] ≟ 0, 1, If[Distance[V, e_1] ≟ 0, 1, 0]]]]) → v2
  49. If(Distance[W, a_1] ≟ 0, 1, If[Distance[W, b_1] ≟ 0, 1, If[Distance[W, c_1] ≟ 0, 1, If[Distance[W, d_1] ≟ 0, 1, If[Distance[W, e_1] ≟ 0, 1, 0]]]]) → w2
  50. If(Distance[Z, a_1] ≟ 0, 1, If[Distance[Z, b_1] ≟ 0, 1, If[Distance[Z, c_1] ≟ 0, 1, If[Distance[Z, d_1] ≟ 0, 1, If[Distance[Z, e_1] ≟ 0, 1, 0]]]]) → z2
  51. If(Distance[AA, a_1] ≟ 0, 1, If[Distance[AA, b_1] ≟ 0, 1, If[Distance[AA, c_1] ≟ 0, 1, If[Distance[AA, d_1] ≟ 0, 1, If[Distance[AA, e_1] ≟ 0, 1, 0]]]]) → aa2
  52. CountIf(x < 1, {x(P_1), x(P_2), x(P_3), x(P_4), x(P_5), y(P_1), y(P_2), y(P_3), y(P_4), y(P_5)}) → zahl1
  53. CountIf(x > 5, {x(P_1), x(P_2), x(P_3), x(P_4), x(P_5), y(P_1), y(P_2), y(P_3), y(P_4), y(P_5)}) → zahl2
  54. If(summe ≟ 0, true, false) → gueltig
  55. If(gueltig ≟ true, false, true) → ungueltig
  56. If(gueltig ≟ true, If[a ≟ true, true, false], false) → gueltig1
  57. Point(Circle[E_1, 1]) → F_1
  58. Intersect(a_1, c_1) → S_1
  59. Intersect(a_1, d_1) → S_2
  60. Intersect(b_1, d_1) → S_3
  61. Intersect(b_1, e_1) → S_4
  62. Intersect(c_1, e_1) → S_5
  63. If(Defined[S_1], true, If[Defined[S_2], true, If[Defined[S_3], true, If[Defined[S_4], true, If[Defined[S_5], true, false]]]]) → notsimple

AI 绘图提示语参考

绘制一个五边形，然后在五边形内部绘制若干点，标记这些点并检查它们是否位于五边形内部。接着，计算每个点到五边形顶点的距离，确定每个点是否与顶点重合。
以上提示语为根据原始绘图命令整理而来，仅供参考。