MathPlotAI Logo MathPlotAI

Proof without words: Area under the cycloid

Proof without words: Area under the cycloid
Loading the math board and drawing, please wait…
2020-01-05 16:46
CalculusIntegralParametric EquationGeometric ConstructionSlider Animation
This GeoGebra example demonstrates, through a dynamic visual proof without words, that the area under one arch of a cycloid (roulette) equals three times the area of its generating circle. The core mathematical concepts involve the parametric equation of the cycloid, definite integrals for area computation, and the correspondence between geometric intuition and algebraic derivation. The construction uses the CurveCartesian command to plot the cycloid, along with Point, Intersect, Polygon, Centroid, Vector, Circle, Segment, Midpoint, Mirror, and Translate tools to partition and transform areas, with a slider animation for dynamic exploration and text objects to annotate the area relationship. Suitable for high school to university-level calculus and analytic geometry instruction, as well as mathematical outreach, this example helps learners visualize the geometric meaning of integration.

GeoGebra Commands & Parameters

  1. CurveCartesian((R * (i - sin(i))), (R * (1 - cos(i))), i, 0, Animation) → b
  2. Point(b) → L
  3. Intersect(xAxis, yAxis) → O
  4. Polygon(D, B, O) → poly2, i, j, l
  5. Centroid(poly2) → M
  6. Vector(M, J) → v
  7. Circle(M, If[Animation < 15, ((Animation - 10) * R / 5), R]) → h
  8. Circle(C, R) → c
  9. Segment(C, A) → d
  10. Polygon(D, E, O) → poly1, o, d_1, e
  11. Segment(D, B) → a
  12. Midpoint(a) → F
  13. Midpoint(O, B) → G
  14. Midpoint(B, E) → H
  15. CurveCartesian((R * (i - sin(i))), (R * (1 - cos(i))), i, 0, If[Animation < (2 * π), Animation, 6.283185307179586]) → f
  16. Mirror(M, a) → M'
  17. Circle(M', If[Animation < 15, ((Animation - 10) * R / 5), R]) → g
  18. Vector(M', I) → u
  19. Translate(g, Vector[(If[Animation < 15, (-10 + Animation) / 5, 1] * u)]) → g'
  20. Circle(M', If[Animation < 15, ((Animation - 10) * R / 5), R]) → p
  21. Translate(h, Vector[(If[Animation < 15, (-10 + Animation) / 5, 1] * v)]) → h'
  22. Translate(p, Vector[(If[Animation < 15, (-10 + Animation) / 5, 1] * u)]) → p'
  23. Circle(M, If[Animation < 15, ((Animation - 10) * R / 5), R]) → k
  24. Translate(k, Vector[(If[Animation < 15, (-10 + Animation) / 5, 1] * v)]) → k'
  25. CurveCartesian((R * (i - sin(i))), (R * (1 - cos(i))), i, 0, If[Animation < (2 * π), Animation, 6.283185307179586]) → m
  26. Point(m) → N
  27. Vector(D, P) → w
  28. Circle(D, If[Animation < 15, ((Animation - 10) * R / 5), R]) → q
  29. Translate(q, Vector[(If[Animation < 15, (-10 + Animation) / 5, 1] * w)]) → q'
  30. Circle(D, If[Animation < 15, ((Animation - 10) * R / 5), R]) → r
  31. Translate(r, Vector[(If[Animation < 15, (-10 + Animation) / 5, 1] * w)]) → r'
  32. Point(p') → Q
  33. Segment(I, Q) → n
  34. Point(k') → S
  35. Segment(J, S) → s
  36. Point(q') → T
  37. Segment(P, T) → a_1
  38. Midpoint(n) → U
  39. Midpoint(s) → V
  40. Midpoint(a_1) → W
  41. Midpoint(d) → Z
  42. CurveCartesian((R * (i - sin(i + π))), ((-R) * (1 - cos(i + π))) + (2 * R), i, (0 * R), If[Animation < (2 * π), Animation, 6.28319]) → z
  43. CurveCartesian((R * (i - sin(i + π))), ((-R) * (1 - cos(i + π))) + (2 * R), i, (0 * R), If[Animation < (2 * π), Animation, 6.28319]) → t
  44. CurveCartesian((R * (i - sin(i + π))), ((-R) * (1 - cos(i + π))) + (2 * R), i, (0 * R), If[Animation < π, Animation, 3.141592653589793]) → b_1
  45. Translate(q, Vector[(If[Animation < 15, (-10 + Animation) / 5, 1] * w)]) → q'_1

AI Drawing Prompt Reference

Draw a dynamic geometric proof showing that the area under one arch of a cycloid equals three times the area of its generating circle. Plot the cycloid and the generating circle, then highlight the area under the cycloid. Use a slider to animate the proof, showing how the cycloid and circle relate geometrically. Partition and transform areas using intersecting lines and circles to visually demonstrate the area relationship.
The prompt above was compiled from the original drawing commands and is for reference only.