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Cavalieri's Principle: Volume Equality of Solids with Equal Cross-Sections

Cavalieri's Principle: Volume Equality of Solids with Equal Cross-Sections
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2016-03-27 13:30
Solid GeometryIntegralGeometric Construction
In collections Geometry - Area & Volume
This GeoGebra example demonstrates Cavalieri's Principle by constructing cubes and prisms and comparing their parallel cross-sectional areas to show that solids with equal height and equal corresponding cross-sectional areas have equal volumes. Core mathematical concepts include solid geometry, volume invariance, cross-sectional area, parallel slicing, and the conceptual bridge to definite integrals and volume formulas. The construction employs GeoGebra commands Cube, Prism, Plane, Intersect, Polygon, Segment, Line, Circle, and PointIn, creating vertices A–H, face objects such as caraABDC and caraBDHE, edge objects such as aristaBD and aristaAB, and auxiliary points I, K, L, M, N, Q, R with segments b, c, d, e, j for dynamic cross-section visualization. It is suitable for high school solid geometry instruction and as an intuitive aid for introducing volume by integration in university calculus.

GeoGebra Commands & Parameters

  1. Cube(B, A, xOyPlane) → a, C, caraACGF, caraCDHG, caraEFGH, aristaBD, aristaAB, aristaAC, aristaCD, aristaBE, aristaDH, aristaEH, D, aristaEF, aristaAF, aristaFG, aristaCG, aristaGH, E, F, G, H, caraABDC, caraBDHE, caraABEF
  2. Intersect(xAxis, yAxis) → I
  3. Plane(K, xOyPlane) → c
  4. PointIn(caraEFGH) → MOVER
  5. Segment(MOVER, A) → b
  6. Intersect(b, c) → L
  7. Line(L, xAxis) → d
  8. Circle(L, 5.7, c) → e
  9. Intersect(d, e) → M, N
  10. Plane(Q, c) → j
  11. Intersect(b, j) → R
  12. Line(R, d) → k
  13. Circle(R, 5.4, j) → p
  14. Intersect(k, p) → S, T
  15. Plane(W, j) → o
  16. Intersect(b, o) → Z
  17. Line(Z, d) → r
  18. Circle(Z, 5.1, o) → s
  19. Intersect(r, s) → A_1, B_1
  20. Plane(E_1, o) → u
  21. Intersect(b, u) → F_1
  22. Line(F_1, d) → d_1
  23. Circle(F_1, 4.8, u) → e_1
  24. Intersect(d_1, e_1) → G_1, H_1
  25. Polygon(F_1, H_1, 4, u) → polígono4, f_1, g_1, h_1, i_1, I_1, J_1
  26. Plane(K_1, u) → v
  27. Intersect(b, v) → L_1
  28. Line(L_1, d) → j_1
  29. Circle(L_1, 4.5, v) → k_1
  30. Intersect(j_1, k_1) → M_1, N_1
  31. Plane(Q_1, v) → w
  32. Intersect(b, w) → R_1
  33. Line(R_1, d) → q_1
  34. Circle(R_1, 4.2, w) → r_1
  35. Intersect(q_1, r_1) → S_1, T_1
  36. Plane(W_1, w) → o_1
  37. Intersect(b, o_1) → Z_1
  38. Line(Z_1, d) → c_2
  39. Circle(Z_1, 3.9, o_1) → d_2
  40. Intersect(c_2, d_2) → A_2, B_2
  41. Plane(E_2, o_1) → u_1
  42. Intersect(b, u_1) → F_2
  43. Line(F_2, d) → i_2
  44. Circle(F_2, 3.6, u_1) → k_2
  45. Intersect(i_2, k_2) → G_2, H_2
  46. Polygon(F_2, H_2, 4, u_1) → polígono8, j_2, l_2, m_2, n_2, I_2, J_2
  47. Plane(K_2, u_1) → v_1
  48. Intersect(b, v_1) → L_2
  49. Line(L_2, d) → p_2
  50. Circle(L_2, 3.3000000000000003, v_1) → q_2
  51. Intersect(p_2, q_2) → M_2, N_2
  52. Plane(Q_2, v_1) → w_1
  53. Intersect(b, w_1) → R_2
  54. Line(R_2, d) → b_3
  55. Circle(R_2, 3, w_1) → c_3
  56. Intersect(b_3, c_3) → S_2, T_2
  57. Plane(W_2, w_1) → o_2
  58. Intersect(b, o_2) → Z_2
  59. Line(A_3, b) → h_3
  60. Line(Z_2, d) → i_3
  61. Circle(Z_2, 2.7, o_2) → k_3
  62. Intersect(i_3, k_3) → B_3, C_3
  63. Plane(F_3, o_2) → u_2
  64. Intersect(b, u_2) → G_3
  65. Line(G_3, d) → p_3
  66. Circle(G_3, 2.4000000000000004, u_2) → q_3
  67. Intersect(p_3, q_3) → H_3, I_3
  68. Polygon(G_3, I_3, 4, u_2) → polígono12, r_3, s_3, t_3, a_4, J_3, K_3
  69. Plane(L_3, u_2) → v_2
  70. Intersect(b, v_2) → M_3
  71. Line(M_3, d) → b_4
  72. Circle(M_3, 2.1, v_2) → c_4
  73. Intersect(b_4, c_4) → N_3, O_3
  74. Plane(R_3, v_2) → w_2
  75. Intersect(b, w_2) → S_3
  76. Line(S_3, d) → h_4
  77. Circle(S_3, 1.7999999999999998, w_2) → k_4
  78. Intersect(h_4, k_4) → T_3, U_3
  79. Plane(Z_3, w_2) → o_3
  80. Intersect(b, o_3) → A_4
  81. Line(A_4, d) → n_4
  82. Circle(A_4, 1.5, o_3) → p_4
  83. Intersect(n_4, p_4) → B_4, C_4
  84. Plane(F_4, o_3) → u_3
  85. Intersect(b, u_3) → G_4
  86. Line(G_4, d) → a_5
  87. Circle(G_4, 1.2000000000000002, u_3) → c_5
  88. Intersect(a_5, c_5) → H_4, I_4
  89. Polygon(G_4, I_4, 4, u_3) → polígono16, b_5, d_5, e_5, f_5, J_4, K_4
  90. Plane(L_4, u_3) → v_3
  91. Intersect(b, v_3) → M_4
  92. Line(M_4, d) → g_5
  93. Circle(M_4, 0.9000000000000004, v_3) → h_5
  94. Intersect(g_5, h_5) → N_4, O_4
  95. Plane(R_4, v_3) → w_3
  96. Intersect(b, w_3) → S_4
  97. Line(S_4, d) → m_5
  98. Circle(S_4, 0.6000000000000005, w_3) → p_5
  99. Intersect(m_5, p_5) → T_4, U_4
  100. Plane(Z_4, w_3) → o_4
  101. Intersect(b, o_4) → A_5
  102. Line(A_5, d) → t_5
  103. Circle(A_5, 0.2999999999999998, o_4) → c_6
  104. Intersect(t_5, c_6) → B_5, C_5
  105. Polygon(B, A, C, D) → polígono20, b_7, a_7, c_7, d_7
  106. Prism(polígono20, 1.2) → u_4, V_2, arista1, arista2, arista3, arista4, arista5, arista6, arista7, arista8, D_3, E_3, P_3, cara1, cara2, cara3, cara4, cara5
  107. Prism(polígono4, 1.2) → u_5, D_2, arista33, arista34, arista35, arista36, arista37, arista38, arista39, arista40, O_2, P_2, U_2, cara21, cara22, cara23, cara24, cara25
  108. Prism(polígono8, 1.2) → g_6, J, arista9, arista10, arista11, arista12, arista13, arista14, arista15, arista16, O, P, U, cara6, cara7, cara8, cara9, cara10
  109. Prism(polígono12, 1.2) → k_6, V, arista17, arista18, arista19, arista20, arista21, arista22, arista23, arista24, C_1, D_1, O_1, cara11, cara12, cara13, cara14, cara15
  110. Prism(polígono16, 1.2) → o_6, P_1, arista25, arista26, arista27, arista28, arista29, arista30, arista31, arista32, U_1, V_1, C_2, cara16, cara17, cara18, cara19, cara20