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Exploring Circle Area Formula

Exploring Circle Area Formula
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2012-11-13 19:00
CircleCalculusGeometric Construction
This example demonstrates the derivation of the circle area formula by cutting a circle into sectors and rearranging them into a shape close to a parallelogram. Key math concepts include circle, sector, limit, and integral calculus. GeoGebra commands like Segment, Rotate, and Sector are used to construct the model, with Slider animation to visualize the limit process. Suitable for high school mathematics education to intuitively understand the origin of the formula.

GeoGebra Commands & Parameters

  1. Segment(E, (x(E) + 2, y(E))) → a
  2. Point(a) → F
  3. Min(x(F) - x(E), 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) - 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. Intersect(p, q, 2) → N
  27. Circle((0, 1), 1) → c_1
  28. Segment((0, 0), (6.283185307179586, 0)) → e
  29. Segment(G, (x(G), y(G) - 1)) → r
  30. 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
  31. 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
  32. CircleArc((0, 1 / (1 - m)), O, L) → s

AI Drawing Prompt Reference

Draw a circle, divide it into multiple sectors, and rearrange these sectors into a shape close to a parallelogram to demonstrate the derivation of the circle area formula. Include an animation slider to visualize the limit process and display key mathematical concepts like circle, sector, limit, and integral calculus.
The prompt above was compiled from the original drawing commands and is for reference only.