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Cavalieri's Principle by Mike May

Cavalieri's Principle by Mike May
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2015-06-22 04:51
Solid GeometryTransformationGeometric Construction
This example intuitively explains Cavalieri's Principle, which states that if two solid figures have equal cross-sectional areas at every level, then they have the same volume. Key concepts include solid geometry and equal areas. The construction uses GeoGebra commands such as Point, Intersect, Segment, etc. Suitable for high school solid geometry education.

GeoGebra Commands & Parameters

  1. Point(xAxis) → B
  2. Point(yAxis) → U_1
  3. Intersect(xAxis, yAxis) → A
  4. Segment(U_1, A) → b_3
  5. Point(b_3) → M_2
  6. Line(A, B) → a
  7. Vector(A, C) → u
  8. Ray(A, C) → b
  9. Line(D, a) → c
  10. Point(c) → G
  11. Circle(G, F) → d
  12. Circle(F, G) → e
  13. OrthogonalLine(G, c) → f
  14. OrthogonalLine(F, c) → g
  15. Intersect(d, f, 2) → H
  16. Intersect(e, g, 2) → I
  17. Intersect(a, f) → J
  18. Intersect(a, g) → K
  19. Translate(J, u4) → J'
  20. Translate(K, u4) → K'
  21. Segment(H, I) → l
  22. Segment(I, F) → m
  23. Segment(H, G) → n
  24. Segment(G, F) → p
  25. Polygon(J, K, K', J') → poly1, j, k, k', j'
  26. Polygon(G, F, I, H) → poly2, g_1, f_1, i, h
  27. Point(c) → L
  28. Point(c) → M
  29. Polygon(L, M, 3) → poly3, q, r, s, N
  30. Dilate(poly3, sqrt(poly2 / poly3), L) → poly3'
  31. Point(c) → O
  32. Point(c) → P
  33. Polygon(O, P, Q, R, S, T, U) → poly4, o, p_1, q_1, r_1, s_1, t, u_1
  34. OrthogonalLine(T, c) → a_1
  35. OrthogonalLine(U, o) → b_1
  36. OrthogonalLine(S, o) → c_1
  37. OrthogonalLine(R, o) → d_1
  38. OrthogonalLine(Q, c) → e_1
  39. Intersect(a_1, b_1) → C_1
  40. OrthogonalLine(P, c) → n_1
  41. OrthogonalLine(O, c) → t_1
  42. Dilate(O, sqrt(poly2 / poly4), O) → O'
  43. Dilate(P, sqrt(poly2 / poly4), O) → P'
  44. Dilate(Q, sqrt(poly2 / poly4), O) → Q'
  45. Dilate(R, sqrt(poly2 / poly4), O) → R'
  46. Dilate(S, sqrt(poly2 / poly4), O) → S'
  47. Dilate(T, sqrt(poly2 / poly4), O) → T'
  48. Dilate(U, sqrt(poly2 / poly4), O) → U'
  49. Polygon(O', P', Q', R', S', T', U') → poly4', o', p', q', r', s', t', u'
  50. OrthogonalLine(O, c) → h_1
  51. OrthogonalLine(P', c) → i_1
  52. OrthogonalLine(Q', c) → j_1
  53. OrthogonalLine(R', c) → k_1
  54. OrthogonalLine(S', c) → l_1
  55. OrthogonalLine(T', c) → m_1
  56. OrthogonalLine(U', c) → a_2
  57. Intersect(h_1, a) → V
  58. Intersect(i_1, a) → W
  59. Intersect(j_1, a) → Z
  60. Intersect(k_1, a) → A_1
  61. Intersect(l_1, a) → B_1
  62. Intersect(m_1, a) → D_1
  63. Intersect(a_2, a) → E_1
  64. Intersect(c, a_2) → F_1
  65. Segment(U', F_1) → b_2
  66. Intersect(m_1, c) → G_1
  67. Intersect(l_1, c) → H_1
  68. Intersect(k_1, c) → I_1
  69. Intersect(j_1, c) → J_1
  70. Segment(U', F_1) → c_2
  71. Segment(T', G_1) → d_2
  72. Segment(S', H_1) → e_2
  73. Segment(R', I_1) → f_2
  74. Segment(Q', J_1) → g_2
  75. Translate(E_1, ub) → K_1
  76. Translate(D_1, ud) → L_1
  77. Translate(B_1, ue) → M_1
  78. Translate(A_1, uf) → N_1
  79. Translate(Z, ug) → Z'
  80. Polygon(K_1, V, W, Z', N_1, M_1, L_1) → poly5, k_2, v, w, z', n_2, m_2, l_2
  81. Dilate(N, sqrt(poly2 / poly3), L) → N'
  82. Dilate(M, sqrt(poly2 / poly3), L) → M'
  83. OrthogonalLine(L, a) → h_2
  84. OrthogonalLine(M', a) → i_2
  85. OrthogonalLine(N', a) → j_2
  86. Intersect(j_2, a) → O_1
  87. Intersect(i_2, a) → P_1
  88. Intersect(h_2, a) → Q_1
  89. Intersect(c, j_2) → R_1
  90. Segment(N', R_1) → p_2
  91. Translate(O_1, up) → S_1
  92. Polygon(Q_1, P_1, S_1) → poly6, s_2, q_2, p_3
  93. Translate(A, u6) → A'
  94. Translate(E, u6) → E'
  95. Point(a) → T_1
  96. Vector(A, T_1) → v_1
  97. Translate(E, v_1) → E'_1
  98. Translate(E', v_1) → E''
  99. Translate(A', v_1) → A''
  100. Midpoint(E'_1, E'') → V_1
  101. Translate(U_1, u6) → W_1
  102. Midpoint(W_1, U_1) → Z_1
  103. Line(Z_1, a) → r_2
  104. Point(r_2) → A_2
  105. Point(r_2) → B_2
  106. Point(r_2) → C_2
  107. Polygon(E, E'_1, E'', E') → poly7, e_3, e'_1, e'', e'
  108. Midpoint(A, A') → D_2
  109. Segment(Z_1, D_2) → t_2
  110. Midpoint(E, E') → E_2
  111. Segment(Z_1, E_2) → a_3
  112. Dilate(Q_1, ratio, A_2) → F_2
  113. Dilate(P_1, ratio, A_2) → G_2
  114. Dilate(S_1, ratio, A_2) → H_2
  115. Dilate(J, ratio, B_2) → J'_1
  116. Dilate(J', ratio, B_2) → J''
  117. Dilate(K', ratio, B_2) → K''
  118. Dilate(K, ratio, B_2) → K'_1
  119. Dilate(V, ratio, C_2) → V'
  120. Dilate(K_1, ratio, C_2) → I_2
  121. Dilate(L_1, ratio, C_2) → J_2
  122. Dilate(M_1, ratio, C_2) → K_2
  123. Dilate(N_1, ratio, C_2) → L_2
  124. Dilate(Z', ratio, C_2) → Z''
  125. Dilate(W, ratio, C_2) → W'
  126. Polygon(F_2, G_2, H_2) → poly8, h_3, f_3, g_3
  127. Polygon(J'_1, K'_1, K'', J'') → poly9, j'_1, k'_1, k'', j''
  128. Polygon(I_2, V', W', Z'', L_2, K_2, J_2) → poly10, i_3, v', w', z'', l_3, k_3, j_3
  129. Segment(Q_1, A_2) → c_3
  130. Segment(S_1, A_2) → d_3
  131. Segment(P_1, A_2) → m_3
  132. Segment(J, B_2) → n_3
  133. Segment(J', B_2) → q_3
  134. Segment(K', B_2) → r_3
  135. Segment(K, B_2) → s_3
  136. Segment(V, C_2) → t_3
  137. Segment(K_1, C_2) → a_4
  138. Segment(L_1, C_2) → b_4
  139. Segment(M_1, C_2) → c_4
  140. Segment(N_1, C_2) → d_4
  141. Segment(Z', C_2) → e_4
  142. Segment(W, C_2) → f_4
  143. Translate(U_1, v_1) → N_2
  144. Translate(W_1, v_1) → O_2
  145. Polygon(U_1, N_2, O_2, W_1) → poly11, u_2, n_4, o_2, w_1
  146. Polygon(A, T_1, A'', A') → poly12, a_5, t_4, a'', a'
  147. Point(e'') → P_2
  148. Point(e'') → Q_2
  149. Point(e'') → R_2
  150. Vector(S_2, T_2) → v_2
  151. Translate(P_2, v_2) → U_2
  152. Translate(Q_2, v_2) → V_2
  153. Translate(R_2, v_2) → W_2
  154. Vector(A, M_2) → w_2
  155. Translate(Z_2, w_2) → B_3
  156. Translate(A_3, w_2) → C_3
  157. Vector(M_2, U_1) → u_3
  158. Translate(B_3, u_3) → D_3
  159. Translate(C_3, u_3) → E_3
  160. Polygon(Z_2, A_3, A'', A') → poly13, z_2, a_6, a''_1, a'_1
  161. Polygon(B_3, C_3, E'', E') → poly14, b_5, c_5, e''_1, e'_2
  162. Polygon(D_3, E_3, O_2, W_1) → poly15, d_5, e_5, o_1, w_3
  163. Midpoint(b_3) → F_3
  164. Vector(G_3, H_3) → v_3
  165. Translate(F_3, v_3) → I_3