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Sphere Surface Area Demonstration

Sphere Surface Area Demonstration
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2018-04-07 03:34
Solid GeometryCalculusParametric EquationGeometric Construction
This example illustrates the geometric representation and calculation principle of the surface area of a sphere. Key math concepts include solid geometry, integral calculus, and parametric equations. The construction utilizes GeoGebra commands/objects such as Intersect, Sphere, Line, Rotate, Circle, Center, Ray, OrthogonalPlane, CircleArc, Point, Plane, Locus, Segment, Mirror, Polygon, Pyramid, and PointIn. It is suitable for high school and above, aiding in teaching solid geometry and spatial visualization skills.

GeoGebra Commands & Parameters

  1. Intersect(zAxis, xOyPlane) → A
  2. Sphere(A, 1) → a
  3. Intersect(zAxis, a) → B, C
  4. Line(A, B) → i
  5. Rotate(C, 120°, xAxis) → C'
  6. Rotate(C', 120°, zAxis) → C''
  7. Circle(C, C', C'') → c_1
  8. Center(c_1) → N
  9. Ray(A, N) → f_1
  10. Intersect(f_1, a) → P, O
  11. OrthogonalPlane(O, f_1) → v
  12. Rotate(C'', 120°, zAxis) → C'''
  13. CircleArc(A, C, C') → c
  14. CircleArc(A, C, C'') → d
  15. CircleArc(A, C', C'') → e
  16. CircleArc(A, C, C''') → f
  17. CircleArc(A, C', C''') → g
  18. CircleArc(A, C'', C''') → h
  19. Circle(B, 1, i) → k
  20. Intersect(j, k) → D, E
  21. Line(D, C') → l
  22. Intersect(l, i) → F
  23. Line(F, C'') → n
  24. Intersect(n, k) → G,
  25. Ray(B, G) → j_1
  26. Line(F, C''') → m
  27. Intersect(m, k) → H,
  28. Point(g) → I
  29. Plane(I, C, A) → o
  30. Intersect(o, k) → J, K
  31. Line(J, I) → p
  32. Intersect(p, i) → L
  33. Locus(M, I) → loc1
  34. Rotate(I, 120°, zAxis) → I'
  35. Rotate(I', 120°, zAxis) → I''
  36. Rotate(M, 120°, zAxis) → M'
  37. Rotate(M', 120°, zAxis) → M''
  38. Locus(M', I) → loc2
  39. Locus(M'', I) → loc3
  40. Segment(C', D) → q
  41. Segment(C''', H) → r
  42. Segment(C'', G) → s
  43. Circle(O, 1, f_1) → d_1
  44. Rotate(d_1, (-(120°)), i) → e_1
  45. Rotate(e_1, (-(120°)), i) → g_1
  46. Ray(A, C') → h_1
  47. Rotate(C', 120°, h_1) → C''_1
  48. Circle(P, 1, f_1) → k_1
  49. Rotate(k_1, (-(120°)), i) → p_1
  50. Rotate(p_1, (-(120°)), i) → q_1
  51. Center(q_1) → Q
  52. Center(p_1) → R
  53. OrthogonalPlane(P, f_1) → u
  54. Ray(B, D) → i_1
  55. Intersect(i_1, u) → S
  56. Ray(B, H) → l_1
  57. Intersect(j_1, v) → U
  58. Intersect(j_1, u) → T
  59. Plane(A, T, U) → w
  60. Mirror(M', w) → M''_1
  61. Locus(M''_1, I) → loc4
  62. Mirror(D, w) → D'
  63. Mirror(G, w) → G'
  64. Polygon(C', C'', C''') → poly1, c''', c', c''
  65. Pyramid(poly1, C) → a_1, faceCC'C'', faceCC''C''', faceCC'C''', edgeCC', edgeCC'', edgeCC'''
  66. PointIn(a) → V