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Volume of Sphere via Steinmetz Solid

Volume of Sphere via Steinmetz Solid
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2013-11-12 11:43
FunctionSolid GeometryCalculusGeometric Construction
This example demonstrates how to derive the volume formula of a sphere by constructing the Steinmetz solid. Key math concepts include volume calculation, solid geometry, and integral concepts. GeoGebra commands and objects used in construction are Sphere, Polygon, Prism, Sequence, Midpoint, and Plane. Suitable for high school students and math enthusiasts learning volume derivation methods.

GeoGebra Commands & Parameters

  1. Sphere(A, 1) → a
  2. Polygon(B, C, D, E) → poly1, b, c, d, e
  3. Prism(poly1, F) → prism1, G, edgeCG, edgeDH, edgeEI, edgeFG, edgeGH, edgeHI, edgeFI, H, I, faceBCGF, faceCDHG, faceDEIH, faceBEIF, faceFGHI, edgeBF
  4. Sequence((sqrt(1 - k^(2)), sqrt(1 - k^(2)), k + 1), k, 0, 1, 0.01) → L1
  5. Sequence((sqrt(1 - k^(2)), (-sqrt(1 - k^(2))), k + 1), k, 0, 1, 0.01) → L2
  6. Sequence(((-sqrt(1 - k^(2))), (-sqrt(1 - k^(2))), k + 1), k, 0, 1, 0.01) → L3
  7. Sequence(((-sqrt(1 - k^(2))), sqrt(1 - k^(2)), k + 1), k, 0, 1, 0.01) → L4
  8. Sequence((sqrt(1 - k^(2)), sqrt(1 - k^(2)), (-k) + 1), k, 0, 1, 0.01) → L5
  9. Sequence((sqrt(1 - k^(2)), (-sqrt(1 - k^(2))), (-k) + 1), k, 0, 1, 0.01) → L6
  10. Sequence(((-sqrt(1 - k^(2))), (-sqrt(1 - k^(2))), (-k) + 1), k, 0, 1, 0.01) → L7
  11. Sequence(((-sqrt(1 - k^(2))), sqrt(1 - k^(2)), (-k) + 1), k, 0, 1, 0.01) → L8
  12. Midpoint(E, B) → M1
  13. Midpoint(C, D) → M2
  14. Midpoint(G, H) → M3
  15. Midpoint(F, I) → M4
  16. Midpoint(B, C) → M5
  17. Midpoint(E, D) → M6
  18. Midpoint(I, H) → M7
  19. Midpoint(F, G) → M8
  20. Midpoint(H, D) → M9
  21. Midpoint(G, C) → M10
  22. Midpoint(F, B) → M11
  23. Midpoint(I, E) → M12
  24. Plane(M1, M2, M3) → i
  25. Plane(M5, M6, M7) → j
  26. Plane(M9, M10, M11) → k
  27. Sequence((0, sqrt(1 - k^(2)), k + 1), k, 0, 1, 0.01) → L9
  28. Sequence((sqrt(1 - k^(2)), 0, k + 1), k, 0, 1, 0.01) → L10
  29. Sequence((0, 0, k + 1), k, 0, 1, 0.01) → L11
  30. Sequence((0 + a2, 1 + a2, k + 1), k, 0, 1, 0.01) → L12
  31. Sequence((1 + a2, 1 + a2, k + 1), k, 0, 1, 0.01) → L13
  32. Sequence((1 + a2, 0 + a2, k + 1), k, 0, 1, 0.01) → L14
  33. Sequence((sqrt(1 - k^(2)) + a2, 0 + a2, k + 1), k, 0, 1, 0.01) → L15
  34. Sequence((sqrt(1 - k^(2)) + a2, sqrt(1 - k^(2)) + a2, k + 1), k, 0, 1, 0.01) → L16
  35. Sequence((0 + a2, sqrt(1 - k^(2)) + a2, k + 1), k, 0, 1, 0.01) → L17
  36. Sequence(Polygon[Element[L12, i], Element[L13, i], Element[L14, i], Element[L15, i], Element[L16, i], Element[L17, i]], i, 1, 101, 1) → list4
  37. Sequence(Polygon[Element[L1, i], Element[L2, i], Element[L3, i], Element[L4, i]], i, 1, 101, 1) → list1
  38. Sequence(Polygon[Element[L5, i], Element[L6, i], Element[L7, i], Element[L8, i]], i, 1, 101, 1) → list2
  39. Sequence(Polygon[Element[L1, i], Element[L2, i], Element[L3, i], Element[L4, i]], i, 1, 101, 104 - (3 * a1)) → list5
  40. Sequence(Polygon[Element[L5, i], Element[L6, i], Element[L7, i], Element[L8, i]], i, 1, 101, 104 - (3 * a1)) → list6
  41. Sequence(Polygon[Element[L11, i], Element[L9, i], Element[L1, i], Element[L10, i]], i, 1, 101, 1) → list3
  42. Polygon(Element[L12, a3], Element[L13, a3], Element[L14, a3], Element[L15, a3], Element[L16, a3], Element[L17, a3]) → poly2, q, r, s_1, t, a_1, b_1
  43. Sequence((1 + a2 - 1.5, 1 + a2 - 1.5, k + 1), k, 0, 1, 0.01) → L18
  44. Sequence(((-k) + a2 - 0.5, 0 + a2 - 0.5, k + 1), k, 0, 1, 0.01) → L19
  45. Sequence(((-k) + a2 - 0.5, (-k) + a2 - 0.5, k + 1), k, 0, 1, 0.01) → L20
  46. Sequence((0 + a2 - 0.5, (-k) + a2 - 0.5, k + 1), k, 0, 1, 0.01) → L21
  47. Sequence(Polygon[Element[L18, i], Element[L19, i], Element[L20, i], Element[L21, i]], i, 1, 101, 1) → list7
  48. Polygon(Element[L18, a3], Element[L19, a3], Element[L20, a3], Element[L21, a3]) → poly3, c_1, d_1, e_1, f_1
  49. Sequence((sqrt(1 - k^(2)), sqrt(1 - k^(2)), k + 1), k, 0, 1, 0.01) → L1_1
  50. Sequence((sqrt(1 - k^(2)), (-sqrt(1 - k^(2))), k + 1), k, 0, 1, 0.01) → L2_1
  51. Sequence(((-sqrt(1 - k^(2))), (-sqrt(1 - k^(2))), k + 1), k, 0, 1, 0.01) → L3_1
  52. Sequence(((-sqrt(1 - k^(2))), sqrt(1 - k^(2)), k + 1), k, 0, 1, 0.01) → L4_1
  53. Sequence(Polygon[Element[L1_1, i], Element[L2_1, i], Element[L3_1, i], Element[L4_1, i]], i, 1, 101, 1) → list1_1
  54. Sequence((sqrt(1 - k^(2)), sqrt(1 - k^(2)), (-k) + 1), k, 0, 1, 0.01) → L5_1
  55. Sequence((sqrt(1 - k^(2)), (-sqrt(1 - k^(2))), (-k) + 1), k, 0, 1, 0.01) → L6_1
  56. Sequence(((-sqrt(1 - k^(2))), (-sqrt(1 - k^(2))), (-k) + 1), k, 0, 1, 0.01) → L7_1
  57. Sequence(((-sqrt(1 - k^(2))), sqrt(1 - k^(2)), (-k) + 1), k, 0, 1, 0.01) → L8_1
  58. Sequence(Polygon[Element[L5_1, i], Element[L6_1, i], Element[L7_1, i], Element[L8_1, i]], i, 1, 101, 1) → list2_1

AI Drawing Prompt Reference

Draw a sphere and demonstrate how to derive the volume formula of a sphere by constructing a Steinmetz solid. Start by creating a unit sphere, then form a polygon and generate its prism. Next, use parametric equations to draw multiple quadrilaterals and construct several planes using these quadrilaterals. Finally, combine these planes with the prism to form a complex geometric solid, illustrating the derivation process of the sphere's volume.
The prompt above was compiled from the original drawing commands and is for reference only.