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Exploring the Area of Circles by garciae

Exploring the Area of Circles by garciae
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2015-02-04 22:13
CircleCalculusTransformationGeometric ConstructionSlider Animation
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, parallelogram, limit, and integral. GeoGebra commands/objects used in construction are Segment, Point, Min, Sequence, Max, Rotate, Sector, Arc, CurveCartesian, Circle, Ray, Midpoint, Intersect, CircleArc. Suitable for high school students learning about the origin of the circle area formula and introductory calculus education.

GeoGebra Commands & Parameters

  1. Segment(E, (x(E) + 3, 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

AI Drawing Prompt Reference

Draw a circle and divide it into multiple sectors, then rearrange these sectors to form a shape close to a parallelogram, demonstrating the derivation of the circle area formula. Include a slider to control the number of sectors, with key elements being the circle, sectors, and the parallelogram formed from rearranging the sectors.
The prompt above was compiled from the original drawing commands and is for reference only.