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Deriving Circle Area into Parallelogram

Deriving Circle Area into Parallelogram
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2017-04-13 01:08
CircleTransformationGeometric Construction
This example demonstrates how to derive the area of a circle into a parallelogram. Key mathematical concepts include circles, parallelograms, and median lines. Using GeoGebra commands such as Segment, Point, Min, Sequence, Max, Rotate, etc., the process of transforming the area of a circle into a parallelogram is constructed. Suitable for high school mathematics education, particularly in coordinate geometry and geometric construction.

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
  34. OrthogonalLine(R, i) → f_1
  35. Intersect(f_1, g) → S
  36. Point(f_1) → T
  37. Segment(S, T) → g_1

AI Drawing Prompt Reference

Draw a geometric construction showing how the area of a circle is transformed into a parallelogram. Begin with a circle, then use a series of rotations and transformations to segment and reconfigure the circle's area into a parallelogram, highlighting key median lines and the transformation process, suitable for high school mathematics education focusing on coordinate geometry and geometric construction.
The prompt above was compiled from the original drawing commands and is for reference only.