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Pyramid Volume Demonstration by Armin Lässer

Pyramid Volume Demonstration by Armin Lässer
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2013-03-23 21:50
VectorSolid GeometryGeometric Construction
In collections Area and Volume (III) S.3 Ch.5 Area and Volume (III)-1 S.3 Ch.5 Area and Volume (III)-2
This example demonstrates the pyramid volume formula using the special case where the base length a is twice the height (a = 2h). It utilizes commands like polygon, vector, and rotate in GeoGebra, suitable for middle school and above students studying solid geometry.

GeoGebra Commands & Parameters

  1. Intersect(yAxis, xAxis) → O
  2. Polygon(A', B'', C', D) → Vieleck1, a', b, c, d
  3. Rotate(A', (-α)°, B'') → A'_2
  4. Ray(B'', A'_2) → e
  5. Circle(B'', v * a') → f
  6. Intersect(f, e) → E,
  7. Vector(B'', E) → u
  8. Translate(A', u) → A'_1
  9. Translate(B'', u) → B'
  10. Translate(C', u) → C'_1
  11. Translate(D, u) → D'
  12. Segment(A'_1, A') → i
  13. Segment(A', C'_1) → j
  14. Segment(D', B'') → k
  15. Intersect(j, k) → F
  16. Vector(F, O) → w
  17. Translate(C'_1, w) → C''
  18. Translate(A', w) → A'_4
  19. Translate(D', w) → D''_2
  20. Translate(B'', w) → B'_2
  21. Translate(D, w) → D'_3
  22. Translate(E, w) → E'
  23. Translate(C', w) → C'_3
  24. Translate(A'_1, w) → I
  25. Vector(O, A) → z
  26. Translate(D'_3, z) → G
  27. Translate(D''_2, z) → H_1
  28. Translate(C'', z) → C'''
  29. Translate(C'_3, z) → J
  30. Mirror(G, O) → G'
  31. Mirror(J, O) → J'
  32. Mirror(C''', O) → K
  33. Mirror(H_1, O) → H'
  34. Line(O, u) → k_1
  35. Vector(O, B) → u_1
  36. Translate(C'', u_1) → C'''_1
  37. Translate(C'_3, u_1) → L
  38. Translate(B'_2, u_1) → M
  39. Translate(E', u_1) → E''
  40. Mirror(A, O) → A'_3
  41. Mirror(L, O) → L'
  42. Mirror(M, O) → M'
  43. Mirror(E'', O) → E'''
  44. Mirror(C'''_1, O) → N
  45. Mirror(B, O) → B'_1
  46. Polygon(G, A, J) → Vieleck2, j_1, g_1, a_1
  47. Polygon(A, C''', J) → Vieleck3, j_2, a_2, c'''
  48. Polygon(H_1, C''', J, G) → Vieleck4, h_1, c'''_1, j_3, g_2
  49. Polygon(E''', N, B'_1) → Vieleck5, b'_1, e''', n
  50. Polygon(M', B'_1, E''') → Vieleck6, e'''_1, m', b'_2
  51. Polygon(B'_1, L', N) → Vieleck7, n_1, b'_3, l'
  52. Polygon(C'''_1, L, M, E'') → Vieleck8, c'''_2, l_1, m_1, e''
  53. Polygon(L, B, M) → Vieleck9, m_2, l_2, b_1
  54. Polygon(K, H', A'_3) → Vieleck10, a'_3, k_2, h'
  55. Polygon(G', H', A'_3) → Vieleck11, a'_2, g', h'_1
  56. Circle(O, v * a) → p
  57. Intersect(p, k_1) → C, P
  58. Vector(O, P) → q
  59. Translate(C'', q) → C'''_3
  60. Translate(D''_2, q) → Q
  61. Translate(E', q) → E''_1
  62. Translate(I, q) → I'
  63. Polygon(Q, P, C'''_3) → Vieleck12, c'''_3, q_1, p_1
  64. Polygon(C'''_3, E''_1, P) → Vieleck13, p_2, c'''_4, e''_1
  65. Polygon(E''_1, I', P) → Vieleck14, p_3, e''_2, i'
  66. Polygon(I', P, Q) → Vieleck15, q_2, i'_1, p_4
  67. Mirror(Q, O) → Q'
  68. Mirror(C'''_3, O) → R
  69. Mirror(I', O) → I''
  70. Mirror(E''_1, O) → S
  71. Polygon(R, Q', I'', S) → Vieleck16, r, q', i'', s
  72. Segment(A, H_1) → t
  73. Segment(I'', C) → c_1
  74. Segment(C, S) → d_1
  75. Segment(C, R) → e_1
  76. Segment(C, Q') → f_1
  77. Segment(B, C'''_1) → i_1
  78. Segment(E'', B) → r_1
  79. Segment(A'_3, J') → s_1
  80. Segment(J', K) → t_1
  81. Segment(J', G') → b_2
  82. Segment(M', L') → c_2
  83. Polygon(E, B'', F) → Vieleck17, f_2, e_2, b''
  84. Polygon(F, A', B'') → Vieleck18, b''_1, f_3, a'_1
  85. Segment(A'_1, E) → g
  86. Segment(A'_1, F) → h_2
  87. Segment(A'_1, B'') → l
  88. Segment(E, A') → m
  89. Intersect(l, m) → T
  90. Segment(F, T) → h

AI Drawing Prompt Reference

Draw a pyramid where the base length is twice the height. Start by drawing the base ABCD, then rotate point A to get A', draw a ray from B' through A', and create a circle centered at B' with radius A'B'. Find the intersection E of the circle and the ray. Draw vector u from E to O, and then translate A', B', C', and D' using vector u. Mirror and rotate these points to form four polygons Vieleck1, Vieleck2, Vieleck3, and Vieleck4, illustrating the pyramid volume formula.
The prompt above was compiled from the original drawing commands and is for reference only.