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Area of a Circle by Bertram Bromfield

Area of a Circle by Bertram Bromfield
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2013-12-24 04:43
CircleFunctionGeometric Construction
This example demonstrates how to calculate the area of a circle using geometric construction. Key mathematical concepts include the properties of circles and the area formula. It utilizes GeoGebra commands such as Circle, Arc, Sector, and Segment. Suitable for middle school and above students learning geometry 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

AI Drawing Prompt Reference

Draw a geometric demonstration of circle area derivation. Start with a horizontal segment and a point on it to control division ratio. Place a center on the vertical axis and generate two point sets: one along an arc and one along a line, dividing the circle into sectors. Translate and rotate these sectors to rearrange them into an approximate rectangle, while keeping an arc and a cycloid as references. Finally, draw a unit circle, a radius segment with its midpoint, and two auxiliary circles to find intersection points, illustrating the relationship between circle area and rectangle area.
The prompt above was compiled from the original drawing commands and is for reference only.