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Copy of Volumes: Exploring Volume through Geometric Transformations

Copy of Volumes: Exploring Volume through Geometric Transformations
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2017-09-17 03:58
PolygonAngleSequenceTransformationGeometric Construction
① This GeoGebra example dynamically demonstrates the projection and replication of 3D volume concepts onto a 2D plane, using rotations, translations, and sequences to construct multiple polygons and compute their areas, thereby visually illustrating the relationship between volume and area. ② Core mathematical concepts include volume, angles, linear equations, triangles, basic calculus, and geometric transformations, involving polygon area calculation, rotational symmetry, and sequence generation. ③ Construction highlights: Commands such as Line, OrthogonalLine, Circle, Intersect, Rotate, Angle, Segment, Midpoint, Polygon, Translate, Sequence, and Point are used to create multiple points, lines, circles, and polygon objects (e.g., A, B, E, F, S', poly1, poly2, poly3), with rotations and translations producing differently positioned figures, and angle and segment measurements aiding analysis. ④ Suitable for upper secondary to early university students, this serves as a bridge between geometry and calculus, and as an interactive tool for teachers to demonstrate volume concepts and geometric transformations.

GeoGebra Commands & Parameters

  1. Line(A, xAxis) → a
  2. OrthogonalLine(A, a) → q_4
  3. Circle(A, 2) → r
  4. Intersect(r, q_4, 2) → E
  5. Intersect(r, a, 2) → B
  6. OrthogonalLine(E, q_4) → s
  7. OrthogonalLine(B, a) → t
  8. Intersect(s, t) → F
  9. Rotate(F, 45*°, E) → S'
  10. Angle(F, E, S') → α
  11. Segment(E, S') → i_1
  12. Segment(E, F) → j_1
  13. Midpoint(j_1) → T
  14. Circle(E, T) → k_1
  15. Intersect(k_1, i_1, 1) → H
  16. Segment(E, H) → p_1
  17. Line(H, j_1) → q_1
  18. Line(F, p_1) → r_1
  19. Intersect(q_1, r_1) → V
  20. Segment(A, E) → s_1
  21. Segment(F, B) → t_1
  22. Line(B, r_1) → a_2
  23. Line(V, s_1) → b_2
  24. Intersect(a_2, b_2) → W
  25. Polygon(A, B, F, E) → poly1, p_2, r_2, s_2, q_2
  26. Polygon(B, W, V, F) → poly2, r_3, w, v, s_3
  27. Polygon(E, F, V, H) → poly3, q_3, s_4, v_1, u
  28. Translate(A, B - A) → A'
  29. Translate(E, B - A) → E'
  30. Sequence(Sequence[Polygon[Translate[A, Vector[k * Vector[A, B] + l * Vector[A, E]]], Translate[B, Vector[k * Vector[A, B] + l * Vector[A, E]]], Translate[F, Vector[k * Vector[A, B] + l * Vector[A, E]]], Translate[E, Vector[k * Vector[A, B] + l * Vector[E, A]]]], k, 0, o], l, 0, 0) → ligne
  31. Sequence(Sequence[Polygon[Translate[E, Vector[l * Vector[A, B] + m * Vector[E, H]]], Translate[F, Vector[l * Vector[A, B] + m * Vector[E, H]]], Translate[V, Vector[l * Vector[A, B] + m * Vector[E, H]]], Translate[H, Vector[l * Vector[A, B] + m * Vector[E, H]]]], l, 0, o], m, 0, q) → liste3
  32. Sequence(Polygon[Translate[B, k * (W - B) + o * (B - A)], Translate[W, k * (W - B) + o * (B - A)], Translate[V, k * (W - B) + o * (B - A)], Translate[F, k * (W - B) + o * (B - A)]], k, 0, q) → ligne2
  33. Sequence(Sequence[Polygon[Translate[A, Vector[k * Vector[B, A] - l * Vector[E, A]]], Translate[B, Vector[k * Vector[B, A] - l * Vector[E, A]]], Translate[F, Vector[k * Vector[B, A] - l * Vector[E, A]]], Translate[E, Vector[k * Vector[B, A] - l * Vector[E, A]]]], k, 0, 0], l, 0, p) → liste
  34. Sequence(Sequence[Polygon[Translate[B, Vector[-l * Vector[E, A] + m * Vector[E, H]]], Translate[W, Vector[-l * Vector[E, A] + m * Vector[E, H]]], Translate[V, Vector[-l * Vector[E, A] + m * Vector[E, H]]], Translate[F, Vector[-l * Vector[E, A] + m * Vector[E, H]]]], l, 0, p], m, 0, 0) → liste2
  35. Translate(E, p * (E - A)) → E'_1
  36. Translate(F, p * (E - A)) → F'
  37. Translate(V, p * (E - A)) → V'
  38. Translate(H, p * (E - A)) → H'
  39. Polygon(E'_1, F', V', H') → Haut, e'_1, f', v', h'
  40. Translate(A, (o + 1) * (B - A)) → Sommet1
  41. Translate(A, (p + 1) * (E - A)) → Sommet2
  42. OrthogonalLine(E'_1, s_1) → b
  43. OrthogonalLine(Sommet1, p_2) → c
  44. Intersect(b, c) → Sommet3
  45. Translate(A, (q + 1) * (H - E)) → Sommet4
  46. OrthogonalLine(Sommet4, p_2) → d
  47. Segment(A, Sommet4) → e
  48. Segment(A, E'_1) → f
  49. Line(E'_1, e) → g
  50. Intersect(d, g) → C
  51. Line(Sommet3, g) → h
  52. Line(C, b) → i
  53. Intersect(h, i) → D
  54. Line(D, d) → j
  55. Line(Sommet1, h) → k
  56. Intersect(j, k) → G
  57. Segment(E'_1, Sommet3) → l
  58. Segment(Sommet3, Sommet1) → m
  59. Segment(Sommet3, D) → n
  60. Segment(D, G) → a_1
  61. Segment(D, C) → b_1
  62. Segment(C, E'_1) → c_1
  63. Segment(F', B) → d_1
  64. Segment(V', V) → e_1
  65. Segment(H', V') → f_1
  66. Segment(A, Sommet1) → g_1
  67. Point(g_1) → I
  68. Segment(I, G) → h_1
  69. Segment(F, E) → l_1
  70. Segment(F, B) → m_1
  71. Line(E, g_1) → n_1
  72. Intersect(n_1, m) → J
  73. Segment(F, J) → c_2
  74. Segment(F', V') → d_2
  75. Segment(B, W) → e_2
  76. Line(J, h_1) → f_2
  77. Intersect(a_1, f_2) → K
  78. Point(h_1) → L
  79. Segment(L, K) → g_2
  80. Segment(J, K) → h_2
  81. Line(H, h_1) → i_2
  82. Line(K, l) → j_2
  83. Intersect(i_2, j_2) → M
  84. Segment(M, K) → k_2
  85. Intersect(i_2, e_1) → N
  86. Segment(N, M) → l_2
  87. Intersect(r_1, j_2) → O
  88. Segment(F, O) → m_2
  89. Sequence(Segment[Translate[F, -k * (A - B)], Translate[O, -k * (A - B)]], k, 0, o) → seg4
  90. Line(N, b_1) → n_2
  91. Intersect(n_2, h_2) → P
  92. Segment(N, P) → t_2
  93. Line(V, t_2) → a_3
  94. Intersect(a_3, h_2) → Q
  95. Line(S', a_3) → b_3
  96. Intersect(b_3, h_2) → R
  97. Segment(V, Q) → c_3
  98. Polygon(F, V, N, M, K, J) → poly4, f_3, v_2, n_3, m_3, k_3, j_3
  99. Polygon(V, F, F', V') → poly5, v_3, f_4, f'_1, v'_1
  100. Segment(N, M) → d_3
  101. Segment(M, K) → e_3
  102. Sequence(Segment[Translate[F, -k * (A - E)], Translate[E, -k * (A - E)]], k, 0, p) → seg_1
  103. Sequence(Segment[Translate[F, k * (B - A)], Translate[B, k * (B - A)]], k, 0, o) → seg_2
  104. Sequence(Segment[Translate[F, -l * (E - H) + o * (B - A)], Translate[B, -l * (E - H) + o * (B - A)]], l, 1, q) → seg10
  105. Sequence(Segment[Translate[N, l * (H - E)], Translate[P, l * (H - E)]], l, -2, q - 3) → seg5
  106. Sequence(Segment[Translate[B, -k * (B - F)], Translate[W, -k * (B - F)]], k, 1, p) → seg_3