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Surface Area of a Sphere by Kriti Singh

Surface Area of a Sphere by Kriti Singh
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2018-06-28 01:40
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates the calculation of a sphere's surface area. Key mathematical concepts include the geometric properties of a sphere and the area formula. Using GeoGebra commands such as Rotate, Segment, and Point, the generation process of the sphere can be dynamically shown. Suitable for high school and above students to learn solid geometry and area calculation.

GeoGebra Commands & Parameters

  1. Line((0, 0, 1), (0, 1, 1)) → c
  2. Sequence(Rotate[(0, 0, 0), ((-((i * 15))))°, c], i, 0, 6) → list1
  3. Rotate(list1, 15°, zAxis) → list1'
  4. Line(Element[list1, 2], Element[list1', 2]) → l1
  5. Line((0, 0, 0), l1) → l0
  6. Sequence(Polygon[Element[list1, i], Element[list1', i], Element[list1', i + 1], Element[list1, i + 1]], i, 1, 6) → list2
  7. Segment(A, A + (0, 0, 1)) → a
  8. Point(a) → P
  9. If(z(P) ≥ 0.5, 0, (2 * (0.5 - z(P)))) → t
  10. If(z(P) ≥ 0.5, (2 * (1 - z(P))), 1) → s
  11. Translate(list2, Vector[(s * Vector[(0, 0, 0), (1, 0, 0)])]) → list2'
  12. Translate(l0, Vector[(s * Vector[(0, 0, 0), (1, 0, 0)])]) → l0'
  13. Sphere((0, 0, 1), 1) → b
  14. Sequence(Pyramid[Element[list2, i], (0, 0, 1)], i, 1, 6) → list3
  15. Translate(list3, Vector[(s * Vector[(0, 0, 0), (1, 0, 0)])]) → list3'
  16. Line(Element[list1, 3], Element[list1', 3]) → l2
  17. Line(Element[list1, 4], Element[list1', 4]) → l3
  18. Line(Element[list1, 5], Element[list1', 5]) → l4
  19. Line(Element[list1, 6], Element[list1', 6]) → l5
  20. Line(Element[list1, 7], Element[list1', 7]) → l6
  21. Rotate(Element[list3', 1], ((t * 7.5))°, l0') → d
  22. Translate(l1, Vector[(s * Vector[(0, 0, 0), (1, 0, 0)])]) → l1'
  23. Translate(l2, Vector[(s * Vector[(0, 0, 0), (1, 0, 0)])]) → l2'
  24. Translate(l3, Vector[(s * Vector[(0, 0, 0), (1, 0, 0)])]) → l3'
  25. Translate(l4, Vector[(s * Vector[(0, 0, 0), (1, 0, 0)])]) → l4'
  26. Rotate(l1', (7.5° * t), l0') → l1''
  27. Rotate(Rotate[Element[list3', 2], ((t * 7.5))°, l0'], ((t * 15))°, l1'') → e
  28. Rotate(Rotate[l2', ((t * 7.5))°, l0'], ((t * 15))°, l1'') → l2''
  29. Rotate(Rotate[Rotate[Element[list3', 3], ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2'') → f
  30. Rotate(Rotate[Rotate[l3', ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2'') → l3''
  31. Rotate(Rotate[Rotate[Rotate[Element[list3', 4], ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], (15° * t), l3'') → g
  32. Rotate(Rotate[Rotate[Rotate[l4', ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], ((t * 15))°, l3'') → l4''
  33. Rotate(Rotate[Rotate[Rotate[Rotate[Element[list3', 5], ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], (15° * t), l3''], (15° * t), l4'') → h
  34. Translate(l5, Vector[(s * Vector[(0, 0, 0), (1, 0, 0)])]) → l5'
  35. Rotate(Rotate[Rotate[Rotate[Rotate[l5', ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], ((t * 15))°, l3''], (15° * t), l4'') → l5''
  36. Rotate(Element[list2', 1], ((t * 7.5))°, l0') → poly1
  37. Rotate(Rotate[Rotate[Rotate[Rotate[Rotate[Element[list3', 6], ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], (15° * t), l3''], (15° * t), l4''], (15° * t), l5'') → i
  38. Rotate(Rotate[Element[list2', 2], ((t * 7.5))°, l0'], ((t * 15))°, l1'') → poly2
  39. Rotate(Rotate[Rotate[Element[list2', 3], ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2'') → poly3
  40. Rotate(Rotate[Rotate[Rotate[Element[list2', 4], ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], (15° * t), l3'') → poly4
  41. Rotate(Rotate[Rotate[Rotate[Rotate[Element[list2', 5], ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], (15° * t), l3''], (15° * t), l4'') → poly5
  42. Rotate(Rotate[Rotate[Rotate[Rotate[Rotate[Element[list2', 6], ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], (15° * t), l3''], (15° * t), l4''], (15° * t), l5'') → poly6
  43. OrthogonalPlane((0, 0, 1), zAxis) → m
  44. Translate(l6, Vector[(s * Vector[(0, 0, 0), (1, 0, 0)])]) → l6'
  45. Rotate(Rotate[Rotate[Rotate[Rotate[Rotate[l6', ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], ((t * 15))°, l3''], (15° * t), l4''], (15° * t), l5'') → l6''
  46. Rotate(Rotate[Rotate[Rotate[Rotate[Rotate[Rotate[m, ((t * 7.5))°, l0'], ((t * 15))°, l1''], ((t * 15))°, l2''], (15° * t), l3''], (15° * t), l4''], (15° * t), l5''], (7.5° * t), l6'') → n
  47. Mirror(i, n) → i'
  48. Mirror(h, n) → h'
  49. Mirror(g, n) → g'
  50. Mirror(f, n) → f'
  51. Mirror(e, n) → e'
  52. Mirror(d, n) → d'
  53. Mirror(poly6, n) → poly6'
  54. Mirror(poly5, n) → poly5'
  55. Mirror(poly4, n) → poly4'
  56. Mirror(poly3, n) → poly3'
  57. Mirror(poly2, n) → poly2'
  58. Mirror(poly1, n) → poly1'
  59. Sequence(Rotate[list3, 15° + ((15° * 22) * i), zAxis], i, 0, 1) → list4
  60. Mirror(list4, m) → list4'
  61. Sequence(Rotate[list2, (15° * i), zAxis], i, 1, 23) → list5
  62. Mirror(list5, m) → list5'