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长方体体积计算 by Néstor Valles-Villarreal

长方体体积计算 by Néstor Valles-Villarreal
几何画板及绘图内容加载中，请稍等……
2012-12-14 06:43
立体几何几何作图
①本范例演示如何通过调整长方体的长、宽、高三个维度，实时计算并可视化其体积变化；②涉及的核心数学概念包括体积计算公式（长×宽×高）、三维空间几何；③作图要点——从下方『构造摘要』提炼用到了 Line（直线）、Polygon（多边形）、Distance（距离）、Rotate（旋转）、Dilate（缩放）等 GeoGebra 命令/对象；④适合初中及以上学生学习立体几何体积计算，也可用于高中数学教学中演示变量间关系。

GeoGebra 命令及参数

  1. Line(AA, BB) → a
  2. OrthogonalLine(CC, a) → b
  3. OrthogonalLine(CC, b) → c
  4. Circle(CC, l) → d
  5. Circle(CC, br) → e
  6. Intersect(d, c) → D, E
  7. Intersect(e, b) → F, G
  8. OrthogonalLine(G, b) → f
  9. OrthogonalLine(D, c) → g
  10. OrthogonalLine(F, b) → h
  11. OrthogonalLine(E, c) → i
  12. Intersect(f, i) → H
  13. Intersect(g, f) → I
  14. Intersect(g, h) → J
  15. Intersect(h, i) → K
  16. Polygon(H, I, J, K) → mangekant1, h_1, i_1, j, k
  17. OrthogonalLine(Rotate, h_1) → m
  18. OrthogonalLine(Rotate, k) → n
  19. Intersect(h_1, m) → M
  20. Intersect(k, n) → N
  21. Distance(Rotate, M) → avstandLM
  22. Distance(Rotate, N) → avstandLN
  23. OrthogonalLine(O, a) → p
  24. OrthogonalLine(O, p) → q
  25. Circle(O, size) → r
  26. Intersect(r, q) → P, Q
  27. Intersect(r, p) → R, S
  28. OrthogonalLine(S, p) → s
  29. OrthogonalLine(P, q) → t
  30. OrthogonalLine(R, p) → a_1
  31. OrthogonalLine(Q, q) → b_1
  32. Intersect(s, b_1) → T
  33. Intersect(s, t) → U
  34. Intersect(a_1, t) → V
  35. Intersect(a_1, b_1) → W
  36. Polygon(U, T, W, V) → mangekant2, u, t_1, w, v
  37. Rotate(U, ho, O) → U'
  38. Rotate(T, ho, O) → T'
  39. Rotate(W, ho, O) → W'
  40. Rotate(V, ho, O) → V'
  41. Polygon(U', T', W', V') → mangekant2', u', t', w', v'
  42. OrthogonalLine(T', q) → c_1
  43. OrthogonalLine(W', q) → d_1
  44. OrthogonalLine(V', q) → e_1
  45. OrthogonalLine(U', q) → f_1
  46. OrthogonalLine(T', q) → g_1
  47. OrthogonalLine(V', q) → j_1
  48. OrthogonalLine(U', q) → k_1
  49. OrthogonalLine(W', q) → l_1
  50. Intersect(g_1, q) → Z
  51. Intersect(l_1, q) → A_1
  52. Intersect(q, k_1) → B_1
  53. Intersect(q, j_1) → C_1
  54. Distance(Z, T') → avstandZT'
  55. Distance(B_1, U') → avstandB1U'
  56. Dilate(T', l1, Z) → T''
  57. Dilate(W', l1, A_1) → W''
  58. Dilate(U', l1, B_1) → U''
  59. Dilate(V', l1, C_1) → V''
  60. Polygon(T'', W'', V'', U'') → mangekant3, t'', w'', v'', u''
  61. Midpoint(W'', V'') → D_1
  62. Midpoint(V'', U'') → E_1
  63. Distance(O, D_1) → avstandOD1
  64. Distance(O, E_1) → avstandOE1
  65. If(((-π < ve) ∧ (ve < (-π) / 2) ∨ (π / 2 < ve) ∧ (ve < π)) ∧ ((-π) / 2 < ho) ∧ (ho < π / 2) ∨ ((-π) / 2 < ve) ∧ (ve < π / 2) ∧ ((-π < ho) ∧ (ho < (-π) / 2) ∨ (π / 2 < ho) ∧ (ho < π)), van / (size - xp) + 3, (-van) / (size - xp) - 3) → ix
  66. Dilate(D_1, ix, O) → F_1
  67. If((0 < ho) ∧ (ho < π) ∧ ((-π) / 2 < ve) ∧ (ve < π / 2) ∨ (-π < ho) ∧ (ho < 0) ∧ ((-π < ve) ∧ (ve < (-π) / 2) ∨ (π / 2 < ve) ∧ (ve < π)), van / (size - yp) + 3, (-van) / (size - yp) - 3) → iy
  68. Dilate(E_1, iy, O) → G_1
  69. If((-π < ve) ∧ (ve < (-π) / 2) ∨ (0 < ve) ∧ (ve < π / 2), (-van) / (size - zp) - 3, van / (size - zp) + 3) → iz
  70. Circle(O, size) → p_1
  71. Circle(O, zp) → q_1
  72. Intersect(q_1, p) → J_1, K_1
  73. Dilate(K_1, iz, O) → H_1
  74. Dilate(D_1, xpos, O) → I_1
  75. Dilate(D_1, xneg, O) → L_1
  76. Dilate(E_1, ypos, O) → M_1
  77. Dilate(E_1, yneg, O) → N_1
  78. Line(M_1, F_1) → m_1
  79. Line(N_1, F_1) → n_1
  80. Line(L_1, G_1) → r_1
  81. Line(I_1, G_1) → s_1
  82. Intersect(n_1, r_1) → O_1
  83. Intersect(m_1, r_1) → P_1
  84. Intersect(s_1, m_1) → Q_1
  85. Intersect(s_1, n_1) → R_1
  86. Polygon(P_1, O_1, R_1, Q_1) → mangekant4, p_2, o_1, r_2, q_2
  87. Dilate(K_1, zpos, O) → S_1
  88. Dilate(K_1, zneg, O) → T_1
  89. Segment(I_1, L_1) → a_2
  90. Segment(N_1, M_1) → b_2
  91. Line(F_1, S_1) → c_2
  92. Line(T_1, F_1) → d_2
  93. Line(S_1, G_1) → e_2
  94. Line(T_1, G_1) → f_2
  95. Line(H_1, N_1) → g_2
  96. Line(H_1, I_1) → h_2
  97. Line(H_1, L_1) → i_2
  98. Line(H_1, M_1) → j_2
  99. Intersect(g_2, e_2) → V_1
  100. Intersect(g_2, f_2) → W_1
  101. Intersect(e_2, j_2) → Z_1
  102. Intersect(f_2, j_2) → A_2
  103. Line(V_1, F_1) → k_2
  104. Line(Z_1, F_1) → l_2
  105. Line(W_1, F_1) → m_2
  106. Line(A_2, F_1) → n_2
  107. Intersect(i_2, c_2) → B_2
  108. Intersect(i_2, d_2) → C_2
  109. Intersect(c_2, h_2) → D_2
  110. Intersect(h_2, d_2) → E_2
  111. Line(B_2, G_1) → s_2
  112. Line(D_2, G_1) → t_2
  113. Line(E_2, G_1) → a_3
  114. Line(C_2, G_1) → b_3
  115. Intersect(k_2, s_2) → F_2
  116. Intersect(s_2, l_2) → G_2
  117. Intersect(t_2, l_2) → H_2
  118. Intersect(t_2, k_2) → I_2
  119. Intersect(m_2, b_3) → J_2
  120. Intersect(m_2, a_3) → K_2
  121. Intersect(a_3, n_2) → L_2
  122. Intersect(n_2, b_3) → M_2
  123. Segment(F_2, G_2) → c_3
  124. Segment(G_2, H_2) → d_3
  125. Segment(H_2, I_2) → e_3
  126. Segment(I_2, F_2) → f_3
  127. Segment(F_2, J_2) → g_3
  128. Segment(I_2, K_2) → h_3
  129. Segment(L_2, H_2) → i_3
  130. Segment(G_2, M_2) → j_3
  131. Segment(M_2, L_2) → k_3
  132. Segment(L_2, K_2) → l_3
  133. Segment(K_2, J_2) → m_3
  134. Segment(J_2, M_2) → n_3
  135. Segment(T_1, S_1) → p_3
  136. Dilate(I_1, enxm, O) → S_2
  137. Line(S_2, G_1) → g_4
  138. Intersect(q_2, g_4) → S_3
  139. Intersect(o_1, g_4) → P_4
  140. Dilate(M_1, enym, O) → N_5
  141. Line(N_5, F_1) → h_6
  142. Intersect(p_2, h_6) → H_6
  143. Intersect(r_2, h_6) → E_7
  144. Dilate(I_1, x1, O) → A
  145. Dilate(I_1, x1m, O) → B
  146. Dilate(M_1, y1, O) → C
  147. Dilate(M_1, y1m, O) → L
  148. Line(L, F_1) → q_3
  149. Line(C, F_1) → r_3
  150. Line(B, G_1) → s_3
  151. Line(A, G_1) → t_3
  152. Intersect(r_3, s_3) → N_2
  153. Intersect(q_3, s_3) → O_2
  154. Intersect(r_3, t_3) → P_2
  155. Intersect(q_3, t_3) → Q_2
  156. Polygon(O_2, Q_2, P_2, N_2) → poly1, o_2, q_4, p_4, n_4
  157. Dilate(I_1, x3, O) → R_2
  158. Dilate(I_1, x2, O) → T_2
  159. Dilate(I_1, x3m, O) → U_2
  160. Dilate(I_1, x2m, O) → V_2
  161. Dilate(M_1, y3, O) → W_2
  162. Dilate(M_1, y2, O) → Z_2
  163. Dilate(M_1, y3m, O) → A_3
  164. Dilate(M_1, y2m, O) → B_3
  165. Dilate(S_1, z1, O) → C_3
  166. Line(V_2, G_1) → a_4
  167. Line(R_2, G_1) → b_4
  168. Line(Z_2, F_1) → c_4
  169. Line(B_3, F_1) → d_4
  170. Intersect(a_4, d_4) → D_3
  171. Intersect(c_4, a_4) → E_3
  172. Intersect(c_4, b_4) → F_3
  173. Intersect(d_4, b_4) → G_3
  174. Polygon(D_3, G_3, F_3, E_3) → poly2, d_5, g_5, f_4, e_4
  175. Line(H_1, T_2) → h_4
  176. Line(H_1, U_2) → i_4
  177. Line(H_1, A_3) → j_4
  178. Line(H_1, W_2) → k_4
  179. Line(C_3, G_1) → l_4
  180. Line(T_2, G_1) → m_4
  181. Line(U_2, G_1) → r_4
  182. Line(A_3, F_1) → s_4
  183. Line(W_2, F_1) → t_4
  184. Line(C_3, F_1) → a_5
  185. Intersect(i_4, a_5) → H_3
  186. Intersect(h_4, a_5) → I_3
  187. Intersect(j_4, l_4) → J_3
  188. Intersect(k_4, l_4) → K_3
  189. Intersect(c_4, m_4) → L_3
  190. Intersect(m_4, d_4) → M_3
  191. Intersect(r_4, c_4) → N_3
  192. Intersect(r_4, d_4) → O_3
  193. Intersect(s_4, a_4) → P_3
  194. Intersect(b_4, s_4) → Q_3
  195. Intersect(t_4, b_4) → R_3
  196. Intersect(t_4, a_4) → T_3
  197. Line(H_1, Q_3) → b_5
  198. Line(H_1, P_3) → c_5
  199. Line(H_1, R_3) → e_5
  200. Line(H_1, T_3) → f_5
  201. Line(H_1, M_3) → h_5
  202. Line(H_1, L_3) → i_5
  203. Line(H_1, O_3) → j_5
  204. Line(H_1, N_3) → k_5
  205. Line(I_3, G_1) → l_5
  206. Line(H_3, G_1) → m_5
  207. Line(J_3, F_1) → n_5
  208. Line(K_3, F_1) → p_5
  209. Intersect(j_5, m_5) → U_3
  210. Intersect(k_5, m_5) → V_3
  211. Polygon(D_3, U_3, V_3, E_3) → poly3, d_6, u_3, v_3, e_6
  212. Intersect(f_5, p_5) → W_3
  213. Intersect(e_5, p_5) → Z_3
  214. Polygon(E_3, W_3, Z_3, F_3) → poly4, e_7, w_3, z_3, f_6
  215. Intersect(c_5, n_5) → A_4
  216. Intersect(b_5, n_5) → B_4
  217. Polygon(A_4, D_3, G_3, B_4) → poly5, a_6, d_7, g_6, b_6
  218. Intersect(i_5, l_5) → C_4
  219. Intersect(h_5, l_5) → D_4
  220. Polygon(G_3, D_4, C_4, F_3) → poly6, g_7, d_8, c_6, f_7
  221. Intersect(a_4, q_3) → E_4
  222. Intersect(b_4, q_3) → F_4
  223. Intersect(t_3, d_4) → G_4
  224. Intersect(t_3, c_4) → H_4
  225. Intersect(r_3, b_4) → I_4
  226. Intersect(r_3, a_4) → J_4
  227. Intersect(d_4, s_3) → K_4
  228. Intersect(s_3, c_4) → L_4
  229. Polygon(L_4, E_3, J_4, N_2) → poly7, l_6, e_8, j_6, n_6
  230. Polygon(E_4, D_3, K_4, O_2) → poly8, e_9, d_9, k_6, o_3
  231. Polygon(I_4, F_3, H_4, P_2) → poly9, i_6, f_8, h_7, p_6
  232. Polygon(Q_2, G_4, G_3, F_4) → poly10, q_5, g_8, g_9, f_9
  233. Segment(O_2, N_2) → r_5
  234. Segment(O_2, Q_2) → s_5
  235. Segment(Q_2, P_2) → t_5
  236. Segment(P_2, N_2) → m_6
  237. Segment(L_4, N_2) → q_6
  238. Segment(N_2, J_4) → r_6
  239. Segment(K_4, O_2) → s_6
  240. Segment(O_2, E_4) → t_6
  241. Segment(F_4, Q_2) → a_7
  242. Segment(Q_2, G_4) → b_7
  243. Segment(H_4, P_2) → c_7
  244. Segment(P_2, I_4) → i_7
  245. Segment(O, C_3) → j_7
  246. Segment(D_3, E_3) → k_7
  247. Segment(E_3, F_3) → l_7
  248. Segment(F_3, G_3) → m_7
  249. Segment(G_3, D_3) → n_7
  250. Segment(D_3, K_4) → p_7
  251. Segment(D_3, E_4) → q_7
  252. Segment(G_3, F_4) → r_7
  253. Segment(G_3, G_4) → s_7
  254. Segment(F_3, H_4) → t_7
  255. Segment(F_3, I_4) → a_8
  256. Segment(E_3, J_4) → b_8
  257. Segment(E_3, L_4) → c_8
  258. Dilate(I_1, hei, O) → M_4
  259. Dilate(S_1, vol, O) → N_4
  260. Line(N_4, F_1) → h_8
  261. Line(H_1, M_4) → i_8
  262. Intersect(h_8, i_8) → O_4
  263. Locus(O_4, height) → loc1
  264. Segment(O, S_1) → j_8
  265. Segment(O, I_1) → k_8
  266. Segment(O_4, N_4) → l_8
  267. Segment(O_4, M_4) → m_8
  268. Segment(O, M_4) → n_8
  269. Segment(T_3, N_2) → p_8
  270. Segment(D_3, U_3) → q_8
  271. Segment(U_3, V_3) → r_8
  272. Segment(V_3, E_3) → s_8
  273. Segment(E_3, W_3) → t_8
  274. Segment(W_3, Z_3) → a_9
  275. Segment(Z_3, F_3) → b_9
  276. Segment(C_4, F_3) → c_9
  277. Segment(C_4, D_4) → h_9
  278. Segment(D_4, G_3) → i_9
  279. Segment(G_3, B_4) → j_9
  280. Segment(B_4, A_4) → k_9
  281. Segment(A_4, D_3) → l_9
  282. Polygon(D_3, E_3, F_3, G_3) → poly11, d_{10}, e_{10}, f_{10}, g_{10}

AI 绘图提示语参考

画一个立方体，通过调整其长、宽、高三个维度，实时显示其体积变化。立方体的底面是一个正方形，通过正方形的对角线构造立方体的侧面，形成六个正方形面。立方体的顶面通过旋转底面并连接对应顶点得到。使用正交线、圆、交点和多边形命令，先构造底面和侧面，再构造顶面，最后通过旋转和连接顶点完成立方体的三维立体效果。
以上提示语为根据原始绘图命令整理而来，仅供参考。