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Exploring the Area of a Circle by Daniela Leone

Exploring the Area of a Circle by Daniela Leone
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2014-10-17 22:51
CircleFunctionGeometric Construction
In collections Deriving area formulas for Geometry Deriving area formulas for Geometry
This example guides students to derive the area formula of a circle by dissecting it into multiple sectors. Key mathematical concepts include the area formula of a circle and sector area calculation. GeoGebra commands used include Segment, Point, Min, Sequence, Max, Rotate, Sector, Arc, CurveCartesian, Circle, Ray, Midpoint, Intersect, CircleArc. Suitable for middle school and above students for geometric construction and area calculation.

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
  33. Segment(Q, R) → g

AI Drawing Prompt Reference

Draw a circle and divide it into multiple sectors to illustrate how the area formula of a circle is derived. Start by drawing a ray and a circle, then segment the circle into several sectors with different colors or fills based on angles. Rearrange these sectors to form a shape resembling a rectangle, where the area is equal to that of the circle. Label key points and lines, such as the center, radius, and sector edges, to clearly demonstrate the derivation process.
The prompt above was compiled from the original drawing commands and is for reference only.