MathPlotAI Logo MathPlotAI

Modifiable Solid: Regular Nonagon Cross Sections Parallel to yAxis

Modifiable Solid: Regular Nonagon Cross Sections Parallel to yAxis
Loading the math board and drawing, please wait…
2019-05-10 18:36
PolygonCoordinate GeometrySolid GeometryCalculusGeometric Construction
In collections GeoGebra 3D & AR: PreCalc & Calculus Resources Cross Sectional Specials (Calculus)
This example demonstrates a modifiable solid where cross sections parallel to the yAxis are regular nonagons. Core mathematical concepts include solid geometry, coordinate geometry, and calculus. Construction highlights the use of GeoGebra commands and objects such as Intersect, Segment, Midpoint, Line, Sphere, Textfield, Rotate, Polygon, CurveCartesian, and Surface. Suitable for high school and above students to explore complex geometric solids and their cross-sectional properties.

GeoGebra Commands & Parameters

  1. Intersect(lower, h, (-10, -3)) → C
  2. Intersect(upper, h, (-10, 13.333333333333332)) → B
  3. Segment(C, B) → q
  4. Intersect(upper, q, (-10, 13.333333333333332)) → G
  5. Midpoint(C, G) → T
  6. Line(T, T + (1, 0, 0)) → l_1
  7. Segment(H, I) → s_1
  8. Midpoint(s_1) → E
  9. Line(E, E + (1, 0, 0)) → r_1
  10. Segment(J, K) → q_1
  11. Midpoint(q_1) → F
  12. Line(F, F + (1, 0, 0)) → c
  13. Line(B, D) → i
  14. Sphere(B, d) → e
  15. Segment(C, B) → p
  16. Textfield(upper) → InputBox1
  17. Textfield(lower) → InputBox2
  18. Textfield(a) → InputBox3
  19. Textfield(b) → InputBox4
  20. Segment(C, G) → k_1
  21. Rotate(J, (2 * pi / 5), Z, c) → J'
  22. Rotate(J', (2 * pi / 5), Z, c) → J''
  23. Rotate(J'', (2 * pi / 5), Z, c) → J'''
  24. Rotate(H, (2 * pi / 5), A_1, r_1) → H'
  25. Rotate(H', (2 * pi / 5), A_1, r_1) → H''
  26. Rotate(H'', (2 * pi / 5), A_1, r_1) → H'''
  27. Rotate(G, (2 * pi / 5), B_1, l_1) → G'
  28. Rotate(G', (2 * pi / 5), B_1, l_1) → G''
  29. Rotate(G'', (2 * pi / 5), B_1, l_1) → G'''
  30. Segment(C, G) → f
  31. Line(C_1, C_1 + (1, 0, 0)) → g
  32. Rotate(G, (2 * pi / 9), C_1, g) → G'_1
  33. Rotate(G'_1, (2 * pi / 9), C_1, g) → D_1
  34. Rotate(D_1, (2 * pi / 9), C_1, g) → E_1
  35. Rotate(E_1, (2 * pi / 9), C_1, g) → F_1
  36. Rotate(F_1, (2 * pi / 9), C_1, g) → G_1
  37. Rotate(G_1, (2 * pi / 9), C_1, g) → H_1
  38. Rotate(H_1, (2 * pi / 9), C_1, g) → I_1
  39. Polygon(C, G, G'_1, D_1, E_1, F_1, G_1, H_1, I_1) → poly1, c_1, g_1, g'_1, d_1, e_1, f_1, g_2, h_2, i_1
  40. Segment(G, G'_1) → j
  41. Segment(G'_1, D_1) → k
  42. Segment(D_1, E_1) → l
  43. Segment(E_1, F_1) → m
  44. Segment(F_1, G_1) → n
  45. Segment(G_1, H_1) → r
  46. Segment(H_1, I_1) → s
  47. Segment(I_1, C) → t
  48. Rotate(J, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → J'_1
  49. Rotate(J'_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → K_1
  50. Rotate(K_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → L_1
  51. Rotate(L_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → M_1
  52. Rotate(M_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → N_1
  53. Rotate(N_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → O_1
  54. Rotate(O_1, (2 * pi / 9), J_1, Line[J_1, J_1 + (1, 0, 0)]) → P_1
  55. Polygon(K, J, J'_1, K_1, L_1, M_1, N_1, O_1, P_1) → poly2, k_2, j_1, j'_1, k_3, l_2, m_1, n_1, o_1, p_1
  56. Segment(J, J'_1) → a_1
  57. Segment(J'_1, K_1) → b_1
  58. Segment(K_1, L_1) → f_2
  59. Segment(L_1, M_1) → i_2
  60. Segment(M_1, N_1) → j_2
  61. Segment(N_1, O_1) → m_2
  62. Segment(O_1, P_1) → n_2
  63. Segment(P_1, K) → p_2
  64. Rotate(H, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → H'_1
  65. Rotate(H'_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → R_1
  66. Rotate(R_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → S_1
  67. Rotate(S_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → T_1
  68. Rotate(T_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → U_1
  69. Rotate(U_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → V_1
  70. Rotate(V_1, (2 * pi / 9), Q_1, Line[Q_1, Q_1 + (1, 0, 0)]) → W_1
  71. Polygon(H, H'_1, R_1, S_1, T_1, U_1, V_1, W_1, I) → poly3, h_3, h'_1, r_2, s_2, t_2, u_1, v_1, w_1, i_3
  72. Segment(I, W_1) → q_2
  73. Segment(W_1, V_1) → a_2
  74. Segment(V_1, U_1) → b_2
  75. Segment(U_1, T_1) → c_2
  76. Segment(T_1, S_1) → d_2
  77. Segment(S_1, R_1) → e_2
  78. Segment(R_1, H'_1) → f_3
  79. Segment(H'_1, H) → g_3
  80. CurveCartesian(t, lower(t) - ((upper(t) - lower(t)) * cos((2 * pi / 9))), ((upper(t) - lower(t)) * sin((2 * pi / 9))), t, a, b) → o
  81. CurveCartesian(t, upper(t) + ((upper(t) - lower(t)) * cos((2 * pi / 9))), ((upper(t) - lower(t)) * sin((2 * pi / 9))), t, a, b) → u
  82. CurveCartesian(t, lower(t) - ((upper(t) - lower(t)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)))), ((upper(t) - lower(t)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)))), t, a, b) → v
  83. CurveCartesian(t, upper(t) + ((upper(t) - lower(t)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)))), ((upper(t) - lower(t)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)))), t, a, b) → w
  84. CurveCartesian(t, lower(t) - ((upper(t) - lower(t)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)) - 0.5)), ((upper(t) - lower(t)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)) + sqrt(3) / 2)), t, a, b) → o_2
  85. CurveCartesian(t, upper(t) + ((upper(t) - lower(t)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)) - 0.5)), ((upper(t) - lower(t)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)) + sqrt(3) / 2)), t, a, b) → u_2
  86. CurveCartesian(t, (0.5 * (upper(t) + lower(t))), ((0.5 * (upper(t) - lower(t))) * (tan((7 * pi / 18)) + 1 / cos((7 * pi / 18)))), t, a, b) → v_2
  87. Surface(u, lower(u) - ((v * (upper(u) - lower(u))) * cos((2 * pi / 9))), ((v * (upper(u) - lower(u))) * sin((2 * pi / 9))), u, a, b, v, 0, 1) → w_2
  88. Surface(u, upper(u) + ((v * (upper(u) - lower(u))) * cos((2 * pi / 9))), ((v * (upper(u) - lower(u))) * sin((2 * pi / 9))), u, a, b, v, 0, 1) → a_3
  89. Surface(u, lower(u) - ((upper(u) - lower(u)) * cos((2 * pi / 9))) - (((upper(u) - lower(u)) * cos((4 * pi / 9))) * v), ((upper(u) - lower(u)) * sin((2 * pi / 9))) + (((upper(u) - lower(u)) * sin((4 * pi / 9))) * v), u, a, b, v, 0, 1) → b_3
  90. Surface(u, upper(u) + ((upper(u) - lower(u)) * cos((2 * pi / 9))) + (((upper(u) - lower(u)) * cos((4 * pi / 9))) * v), ((upper(u) - lower(u)) * sin((2 * pi / 9))) + (((upper(u) - lower(u)) * sin((4 * pi / 9))) * v), u, a, b, v, 0, 1) → c_3
  91. Surface(u, lower(u) - ((upper(u) - lower(u)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)))) + ((0.5 * (upper(u) - lower(u))) * v), ((upper(u) - lower(u)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)) + (sqrt(3) / 2 * v))), u, a, b, v, 0, 1) → d_3
  92. Surface(u, upper(u) + ((upper(u) - lower(u)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)) - (0.5 * v))), ((upper(u) - lower(u)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)) + (sqrt(3) / 2 * v))), u, a, b, v, 0, 1) → e_3
  93. Surface(u, lower(u) - ((upper(u) - lower(u)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)) - 0.5 - (cos(pi / 9) * v))), ((upper(u) - lower(u)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)) + sqrt(3) / 2 + (v * sin(pi / 9)))), u, a, b, v, 0, 1) → l_3
  94. Surface(u, upper(u) + ((upper(u) - lower(u)) * (cos((2 * pi / 9)) + cos((4 * pi / 9)) - 0.5 - (cos(pi / 9) * v))), ((upper(u) - lower(u)) * (sin((2 * pi / 9)) + sin((4 * pi / 9)) + sqrt(3) / 2 + (v * sin(pi / 9)))), u, a, b, v, 0, 1) → j_3