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Deriving the Circumference and Area of a Circle

Deriving the Circumference and Area of a Circle
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2013-02-24 14:28
CirclePolygonCalculus
In collections Area and circumference of a circle
This example demonstrates the derivation of the formulas for the circumference and area of a circle using the regular polygon method and a modified onion method. It utilizes commands such as Sequence, Polygon, Circle, Point, and Segment in GeoGebra, suitable for students in middle school and above studying geometry and calculus.

GeoGebra Commands & Parameters

  1. Sequence((sin(2 * π * k / n), -cos(2 * π * k / n)), k, 0, n, 1) → vertices
  2. Polygon(vertices) → regpoly, , , ,
  3. Segment((0, 0), Midpoint[Element[vertices, 1], Element[vertices, 2]]) → a
  4. Perimeter(regpoly) → p
  5. Sequence(Segment[(0, 0), Element[vertices, k]], k, 1, n) → trisides
  6. Area(regpoly) → A
  7. Circle(O, 1) → c
  8. Segment(O, B) → r
  9. Sequence(Circle[O, j / k], j, 1, k, 1) → circles_0
  10. Sequence(O + (0, (-j) / k), j, 1, k, 1) → initial
  11. Zip(Circle[C, P], P, initial) → circles
  12. Zip(Rotate[P, 360° * Distance[O, P] / Radius[S], C], P, initial, S, circles) → final
  13. Zip(Ray[C, P], P, final) → rays
  14. Sequence(Intersect[Element[circles, j], Element[rays, j + 1]], j, 1, k - 1, 1) → final_2
  15. Zip(CircleArc[C, P, P'], P, initial, P', final_2) → arcs
  16. CircleArc(C, O, Rotate[O, 360° * Distance[O, Element[initial, 1]] / Radius[Element[circles, 1]], C]) → arc_0
  17. Sequence(Segment[Element[final, j + 1], Element[final_2, j]], j, 1, k - 1, 1) → endseg
  18. CircleArc(C, B, Rotate[B, 360° * Distance[O, B] / Radius[Element[circles, k]], C]) → arc_k
  19. Sequence(Element[initial, j] + (2 * π * Radius[Element[circles_0, j + 1]], 0), j, 1, k - 1, 1) → finalh_2
  20. Sequence(Segment[Element[initial, j], Element[finalh_2, j]], j, 1, k - 1, 1) → bases
  21. Segment(O, O + (2 * π * Radius[Element[circles_0, 1]], 0)) → base_0
  22. Segment(B, B + (2 * π * Radius[Element[circles_0, k]], 0)) → base_k
  23. Zip(P + (2 * π * Radius[S], 0), P, initial, S, circles_0) → finalh
  24. Sequence(Segment[Element[finalh_2, j - 1], Element[finalh, j]], j, 2, k, 1) → heights
  25. Segment(finalh_0, Element[finalh, 1]) → height_0
  26. Intersect(arc_0, Element[rays, 1]) → final_0,
  27. Segment(final_0, Element[final, 1]) → endseg_0
  28. Point(Circle[D, 6.283185307179586]) → E
  29. Segment(D, E) → e
  30. Sequence(1 / k * 2 * π * Radius[Element[circles_0, j]], j, 1, k) → areas
  31. Sum(areas) → area

AI Drawing Prompt Reference

Draw a circle and demonstrate the derivation of its circumference and area. Start by approximating the circle with a regular polygon, then use rotations and divisions to approximate the circumference and area. Next, construct a series of arcs and segments to illustrate the limit definitions of the circumference and area. Finally, draw the diameter and radius of the circle, and show the area formula.
The prompt above was compiled from the original drawing commands and is for reference only.