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Exploring the Area of a Circle by tconrath

Exploring the Area of a Circle by tconrath
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2011-10-26 07:12
CircleCalculusGeometric Construction
In collections Area of Surfaces
By cutting a circle into sectors and rearranging them to form a figure close to a parallelogram, this example explores the formula for the area of a circle. It involves the geometric properties of circles and area calculation, using GeoGebra commands such as segment, sector, and arc. Suitable for middle school and above students to learn the method of calculating the area of a circle.

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