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圆面积公式探索 by Aline Lemmertz

圆面积公式探索 by Aline Lemmertz
几何画板及绘图内容加载中，请稍等……
2017-05-20 21:02
圆三角函数几何作图
①本范例通过将圆切割成扇形并重新排列成近似平行四边形的图形，演示圆面积公式的推导过程；②涉及的核心数学概念包括圆、扇形、面积计算、几何变换；③作图要点提炼自构造摘要，使用了Segment, Point, Rotate, Sector, Circle等GeoGebra命令和对象；④适合初中及以上学生，用于直观理解圆面积公式的来源和几何意义。

GeoGebra 命令及参数

  1. Segment(E, (x(E) + 4, y(E))) → a
  2. Point(a) → F
  3. Min((x(F) - x(E)) / (a / 2), 1) → k
  4. Point(yAxis) → O
  5. Sequence(If[k < 1, (x(O) + ((l * sin((i * φ))) * cos((i * φ)) / sin(φ)), y(O) + ((l * sin((i * φ))) * sin((i * φ)) / sin(φ))), (x(O) + (l * i), y(O))], i, 0, (2 * n)) → list2
  6. Sequence((x(O) + sin((i * θ)), y(O) + 1 - cos((i * θ))), i, 0, (2 * n)) → list1
  7. Max((x(F) - x(E)) / (a / 2) - 1, 0) → k'
  8. Rotate((x(O) + 0, y(O) + 1), ((-k) * θ / 2), O) → A
  9. Rotate(O, ((-k) * θ / 2), O) → O'
  10. Rotate(Element[list1, 1], ((-k) * θ / 2), O) → B
  11. Sector(Rotate[Circle[(x(O) + 0, y(O) + 1), O], ((-k) * θ / 2), O], Rotate[O, ((-k) * θ / 2), O], Rotate[Element[list1, 1], ((-k) * θ / 2), O]) → c
  12. Rotate((x(O) + 0, y(O) + 1), ((-k) * θ / 2), O) → C
  13. Rotate(O, ((-k) * θ / 2), O) → O'_1
  14. Rotate(Element[list1, 1], ((-k) * θ / 2), O) → D
  15. Arc(Rotate[Circle[(x(O) + 0, y(O) + 1), O], ((-k) * θ / 2), O], O'_1, D) → d
  16. CurveCartesian(t - sin(t), 1 - cos(t), t, 0, 6.28319) → b
  17. Circle((0, 1), 1) → h
  18. Ray((0, 0), (1, 0)) → i
  19. Point(Segment[(0, 1), (6.28319, 1)]) → H
  20. Segment(H, (x(H), 0)) → j
  21. Midpoint(j) → I
  22. Midpoint(a) → J
  23. Circle(L, 1 / (1 - m)) → p
  24. Circle((0, 0), 1 / (1 - m)) → q
  25. Intersect(p, q, 1) → M
  26. Circle((0, 1), 1) → c_1
  27. Segment((0, 0), (6.283185307179586, 0)) → e
  28. Segment(G, (x(G), y(G) - 1)) → r
  29. Sequence(Sector[Rotate[Translate[Circle[(x(O) + 0, y(O) + 1), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], Rotate[Translate[Element[list1, i], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], Rotate[Translate[Element[list1, i + 1], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]]], i, 1, n) → list3
  30. Sequence(Sector[Rotate[Rotate[Translate[Circle[(x(O) + 0, y(O) + 1), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O) + (n * l) - sin(θ / 2) / 2, y(O) + cos(θ / 2) / 2)], Rotate[Rotate[Translate[Element[list1, i], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O) + (n * l) - sin(θ / 2) / 2, y(O) + cos(θ / 2) / 2)], Rotate[Rotate[Translate[Element[list1, i + 1], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O) + (n * l) - sin(θ / 2) / 2, y(O) + cos(θ / 2) / 2)]], i, n + 1, (2 * n)) → list4
  31. CircleArc((0, 1 / (1 - m)), O, L) → s