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Surface Area by Pappus Theorem

Surface Area by Pappus Theorem
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2020-01-22 20:40
Solid GeometryCalculusTransformationGeometric Construction
This example demonstrates the calculation of surface area of solids of revolution using Pappus' Theorem. Key math concepts include Pappus' Theorem, surface area of solids, and geometric transformations. Construction involves commands like Sequence, Polygon, Centroid, Circle, Conic, Segment, Midpoint, Vector, Line, CircleArc, Surface, Rotate, CircumcircleArc to visualize and apply the theorem. Suitable for high school students and math enthusiasts studying solid geometry and applications of Pappus' Theorem.

GeoGebra Commands & Parameters

  1. Sequence(Rotate[A + (r, 0), i, A], i, ((-pi)) / 2, pi / 2, (2 * pi / n)) → l1
  2. Polygon(l1) → poly1
  3. Centroid(poly1) → B
  4. Polygon(E, G, F, D) → q1, e, g, f, d
  5. Polygon(J, K, I) → t1, i, j, k
  6. Circle(A, r) → c
  7. Conic(K, D_1, Z, C_1, P) → h
  8. Conic(F, E_1, B_1, A_1, Q) → p
  9. Conic(S, G, J_1, W, V) → q
  10. Conic(T, H_1, G_1, U, I_1) → r_2
  11. Segment(V, A_1) → l
  12. Segment(J, C_1) → m
  13. Centroid(q1) → K_1
  14. Centroid(t1) → L_1
  15. Midpoint(N_1, O_1) → T_1
  16. Midpoint(P_1, Q_1) → U_1
  17. Midpoint(R_1, S_1) → V_1
  18. Vector(T_1, O_1) → u
  19. Vector(T_1, N_1) → v
  20. Vector(U_1, Q_1) → w
  21. Vector(U_1, P_1) → a
  22. Vector(V_1, S_1) → b
  23. Vector(V_1, R_1) → s
  24. Segment(K, O_1) → t
  25. Segment(I, N_1) → f_1
  26. Segment(D, P_1) → g_1
  27. Segment(F, Q_1) → h_1
  28. Segment(M, R_1) → i_1
  29. Segment(H_1, S_1) → j_1
  30. Segment(L_1, W_1) → k_1
  31. Midpoint(P, W_1) → Z_1
  32. Vector(P, W_1) → u_1
  33. Vector(Z_1, P) → v_1
  34. Midpoint(Q, A_2) → B_2
  35. Midpoint(A, B) → C_2
  36. Midpoint(M_1, L_1) → D_2
  37. Midpoint(C, K_1) → E_2
  38. Segment(M_1, L_1) → m_1
  39. Line(N, zAxis) → p_1
  40. Line(F_1, zAxis) → q_1
  41. Line(F_2, zAxis) → r_1
  42. Circle(N, r, p_1) → c_1
  43. Polygon(G_2, N, H_2) → t2, h_2, g_2, n_2
  44. Circle(F_1, r, q1) → d_1
  45. Circle(J_2, r, q1) → e_1
  46. Polygon(J_2, L_2, K_2, F_1) → q2, j_2, l_2, k_2, s_1
  47. Line(N_2, yAxis) → t_1
  48. Sequence(Rotate[N_2 + (r, 0), i, t_1], i, ((-pi)) / 2, pi / 2, (2 * pi / n)) → l2
  49. Polygon(l2) → poly2
  50. Circle(N_2, r, r_1) → c_2
  51. CircleArc(A, M, L) → d_2
  52. Segment(I, J) → l_1
  53. Segment(E, D) → n_1
  54. Segment(M, L) → f_2
  55. Segment(H, I_2) → i_2
  56. Segment(A, O_2) → p_2
  57. Segment(C, M_2) → m_2
  58. Surface(n_2, Rotation, p_1) → a_1
  59. Rotate(H_2, Rotation, p_1) → P_2
  60. Rotate(G_2, Rotation, p_1) → Q_2
  61. Segment(P_2, Q_2) → q_2
  62. Circle(N, r, p_1) → e_2
  63. Surface(l_2, Rotation, q_1) → b_1
  64. Rotate(L_2, Rotation, q_1) → R_2
  65. Rotate(K_2, Rotation, q_1) → S_2
  66. Segment(R_2, S_2) → s_2
  67. CircumcircleArc(U_2, T_2, F_2) → t_2
  68. Rotate(t_2, Rotation, r_1) → c_3
  69. Surface(t_2, Rotation, r_1) → w_1
  70. Segment(U_2, F_2) → a_2
  71. Segment(J_2, F_1) → b_2
  72. Segment(N, G_2) → f_3
  73. Sequence(I + ((r * cos(i)), (r / 3 * sin(i))), i, 0, (2 * pi), (2 * pi / 20)) → l3
  74. Segment(J, Element[l3, dd]) → g_3
  75. Sequence(D + ((r * cos(i)), (r / 3 * sin(i))), i, 0, (2 * pi), (2 * pi / 20)) → l4
  76. Sequence(E + ((r * cos(i)), (r / 3 * sin(i))), i, 0, (2 * pi), (2 * pi / 20)) → l5
  77. Segment(Element[l4, dd], Element[l5, dd]) → h_3
  78. Sequence(A + ((r * cos(i)), (r / 3 * sin(i))), i, 0, (2 * pi), (2 * pi / 20)) → l6
  79. CircumcircleArc(L, Element[l6, dd], M) → d_3

AI Drawing Prompt Reference

Visualize the calculation of surface area for solids of revolution using Pappus' Theorem. Start by drawing a rotating curve, then generate the solid through rotations and geometric transformations, highlighting key points like centroids and vertices. Include auxiliary lines and circles to demonstrate the theorem's application, and present the surface and volume calculations of the solid.
The prompt above was compiled from the original drawing commands and is for reference only.