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Wordless Proof: Area of the Sphere

Wordless Proof: Area of the Sphere
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2020-01-10 16:33
Solid GeometryCalculusGeometric ConstructionGraphing
This example dynamically demonstrates the proof of the sphere's surface area formula. It involves core math concepts like sphere, cylinder volume, calculus, and integration. Constructed using GeoGebra commands such as CurveCartesian, Surface, Point, Line, Vector, Translate, Segment, Midpoint, Rotate, Circle, Polygon. Suitable for high school students and math enthusiasts learning calculus and solid geometry.

GeoGebra Commands & Parameters

  1. CurveCartesian(((0 * (-R)) + (R * sin((((-k)) / 4 * π))), 0, (0 * (((k / 4 * 2) * R) - R)) + ((-R) * cos(((-k) * π / 4)))), k, 0, 4) → e
  2. CurveCartesian(((R * sin(((-t) * π))), 0, ((-R) * cos(((-t) * π)))), t, 0, 1) → Sph
  3. CurveCartesian(((Animation * (-R)) + ((1 - Animation) * (R * sin((((-k)) / 4 * π)))), 0, (Animation * (((k / 4 * 2) * R) - R)) + ((1 - Animation) * ((-R) * cos(((-k) * π / 4))))), k, 0, 4) → SphLine
  4. Surface(SphLine, 360°, zAxis) → SPH
  5. CurveCartesian(((R * cos(((2 * (-t)) * π))), (R * sin(((2 * (-t)) * π)))), t, 0, 1) → Circl2
  6. CurveCartesian((((Animation - 1) * (-R)) + ((2 - Animation) * (R * cos(((2 * (-k)) * π)))), ((Animation - 1) * ((((k * 2) * π) * R) - (π * R))) + ((2 - Animation) * (R * sin(((2 * (-k)) * π))))), k, 0, 1) → SphLine2
  7. Surface(((Animation - 1) * (-R)) + ((2 - Animation) * (R * cos(((2 * ((-u)) / 8) * π)))), ((Animation - 1) * ((((u / 8 * 2) * π) * R) - (π * R))) + ((2 - Animation) * (R * sin(((2 * ((-u)) / 8) * π)))), ((v / 4 * 2) * R) - R, u, 0, 8, v, 0, 4) → SPH2
  8. CurveCartesian(((-R), (((t * 2) * π) * R) - (π * R), R), t, 0, 1) → Lin2
  9. CurveCartesian(((-R), (-π), ((t * 2) * R) - R), t, 0, 1) → Line
  10. Point(Lin2) → A
  11. Point(Line) → B
  12. Surface(x(C) + (f(s) * cos((-t))), y(C) + (f(s) * sin(t)), z(C) - f_2(s), s, -1.5708, π, t, 0, (2 * π)) → b
  13. Line(C, zAxis) → g
  14. Vector((0, 0, 0), C) → u
  15. Surface(x(D) + ((R * cos(i)) * cos(j)), y(D) + ((R * cos(i)) * sin(j)), z(D) + (R * sin(i)), i, 0, (2 * π), j, 0, (2 * π)) → c
  16. Line(D, zAxis) → h
  17. Translate(e, Vector[(0, 0, 0), D]) → Sph'
  18. Surface(Sph', 360°, h) → d
  19. CurveCartesian(((If[Animation < 1, Animation, 1] * (-R)) + (((1 - If[Animation < 1, Animation, 1]) * R) * sin((((-k)) / 4 * π))), 0, (If[Animation < 1, Animation, 1] * (((k / 4 * 2) * R) - R)) + ((1 - If[Animation < 1, Animation, 1]) * ((-R) * cos(((-k) * π / 4))))), k, 0, 4) → i
  20. Translate(i, Vector[(0, 0, 0), C]) → i'
  21. Surface(i', 360°, g) → SPH'
  22. Vector(E, F) → v
  23. Vector(G, H) → w
  24. Segment(D, I) → j
  25. Point(i') → L
  26. Segment(J, L) → k
  27. Segment(J, K) → l
  28. Midpoint(k) → M
  29. Midpoint(J, C) → N
  30. Midpoint(C, K) → O
  31. Midpoint(D, I) → P
  32. Rotate(L, 90°, l) → L'
  33. Rotate(L, (-(90°)), l) → L'_1
  34. Circle(L', L, L'_1) → p
  35. Polygon(AA, BB, CC, DD) → q1, aa, bb, cc, dd
  36. Vector(Q, S) → a

AI Drawing Prompt Reference

Create a 3D geometric illustration demonstrating the surface area formula of a sphere. Show a sphere formed by rotating a semicircle around its axis, alongside a tangent cylinder. Animate the unfolding of a portion of the sphere's surface onto the cylinder's lateral area, visually proving their areas are equal. Include auxiliary circles, line segments, vectors, and labeled points to indicate correspondences. Present the scene in a three-dimensional coordinate system, emphasizing the sphere, cylinder, and the unfolding process.
The prompt above was compiled from the original drawing commands and is for reference only.