积极 AI Logo 积极 AI

无字证明：金字塔体积

无字证明：金字塔体积
几何画板及绘图内容加载中，请稍等……
2019-12-27 05:09
向量解析几何立体几何几何作图
①本范例通过动态演示证明金字塔体积公式；②涉及核心数学概念：体积、几何体、比例关系；③构造要点：运用了 Midpoint（中点）、Polygon（多边形）、Line（直线）、Vector（向量）、If（条件）、Sort（排序）、Segment（线段）、Intersect（交点）、RotateText（旋转文本）、Translate（平移）等 GeoGebra 命令；④适合高中及以上学生及数学爱好者学习立体几何体积计算方法。

GeoGebra 命令及参数

  1. Midpoint(Z_1, N) → I_2
  2. Polygon(W_1, Z_1, Q, N) → poly18, w_2, z_2, q_1, n_2
  3. Polygon(R, S, Q, N) → poly7, r, s, q, n_1
  4. Line(I_2, s) → b_2
  5. Vector(C, X) → u
  6. Vector(C, Y) → v
  7. Vector(C, Z) → w
  8. If(Animation < 0, 0, If[Animation > 1.5, 1.5, Animation]) → m
  9. Sort(CellRange(J1,J6,"numeric")) → liste1
  10. If(Angle[E1, F1, G1] < 180°, Polygon[CellRange(E1,H1,"point")]) → L1
  11. If(Angle[E2, F2, G2] < 180°, Polygon[CellRange(E2,H2,"point")]) → L2
  12. If(Angle[E3, F3, G3] < 180°, Polygon[CellRange(E3,H3,"point")]) → L3
  13. If(Angle[E4, F4, G4] < 180°, Polygon[CellRange(E4,H4,"point")]) → L4
  14. If(Angle[E5, F5, G5] < 180°, Polygon[CellRange(E5,H5,"point")]) → L5
  15. If(Angle[E6, F6, G6] < 180°, Polygon[CellRange(E6,H6,"point")]) → L6
  16. If(Angle[E1, H1, I1] < 180° ∧ m > 0, Polygon[E1, H1, I1]) → M1
  17. If(Angle[H1, G1, I1] < 180° ∧ m > 0, Polygon[H1, G1, I1]) → N1
  18. If(Angle[G1, F1, I1] < 180° ∧ m > 0, Polygon[G1, F1, I1]) → O1
  19. If(Angle[F1, E1, I1] < 180° ∧ m > 0, Polygon[F1, E1, I1]) → P1
  20. If($K1 ≟ 1, M1) → Q1
  21. If($K1 ≟ 1, N1) → R1
  22. If($K1 ≟ 1, O1) → S1
  23. If($K1 ≟ 1, P1) → T1
  24. If($K1 ≟ 2, M1) → U1
  25. If($K1 ≟ 2, N1) → V1
  26. If($K1 ≟ 2, O1) → W1
  27. If($K1 ≟ 2, P1) → X1
  28. If($K1 ≟ 3, M1) → Y1
  29. If($K1 ≟ 3, N1) → Z1
  30. If($K1 ≟ 3, O1) → AA1
  31. If($K1 ≟ 3, P1) → AB1
  32. If($K1 ≟ 4, M1) → AC1
  33. If($K1 ≟ 4, N1) → AD1
  34. If($K1 ≟ 4, O1) → AE1
  35. If($K1 ≟ 4, P1) → AF1
  36. If($K1 ≟ 5, M1) → AG1
  37. If($K1 ≟ 5, N1) → AH1
  38. If($K1 ≟ 5, O1) → AI1
  39. If($K1 ≟ 5, P1) → AJ1
  40. If(Angle[E2, H2, I2] < 180° ∧ m > 0, Polygon[E2, H2, I2]) → M2
  41. If(Angle[H2, G2, I2] < 180° ∧ m > 0, Polygon[H2, G2, I2]) → N2
  42. If(Angle[G2, F2, I2] < 180° ∧ m > 0, Polygon[G2, F2, I2]) → O2
  43. If(Angle[F2, E2, I2] < 180° ∧ m > 0, Polygon[F2, E2, I2]) → P2
  44. If($K2 ≟ 1, M2) → Q2
  45. If($K2 ≟ 1, N2) → R2
  46. If($K2 ≟ 1, O2) → S2
  47. If($K2 ≟ 1, P2) → T2
  48. If($K2 ≟ 2, M2) → U2
  49. If($K2 ≟ 2, N2) → V2
  50. If($K2 ≟ 2, O2) → W2
  51. If($K2 ≟ 2, P2) → X2
  52. If($K2 ≟ 3, M2) → Y2
  53. If($K2 ≟ 3, N2) → Z2
  54. If($K2 ≟ 3, O2) → AA2
  55. If($K2 ≟ 3, P2) → AB2
  56. If($K2 ≟ 4, M2) → AC2
  57. If($K2 ≟ 4, N2) → AD2
  58. If($K2 ≟ 4, O2) → AE2
  59. If($K2 ≟ 4, P2) → AF2
  60. If($K2 ≟ 5, M2) → AG2
  61. If($K2 ≟ 5, N2) → AH2
  62. If($K2 ≟ 5, O2) → AI2
  63. If($K2 ≟ 5, P2) → AJ2
  64. If(Angle[E3, H3, I3] < 180° ∧ m > 0, Polygon[E3, H3, I3]) → M3
  65. If(Angle[H3, G3, I3] < 180° ∧ m > 0, Polygon[H3, G3, I3]) → N3
  66. If(Angle[G3, F3, I3] < 180° ∧ m > 0, Polygon[G3, F3, I3]) → O3
  67. If(Angle[F3, E3, I3] < 180° ∧ m > 0, Polygon[F3, E3, I3]) → P3
  68. If($K3 ≟ 1, M3) → Q3
  69. If($K3 ≟ 1, N3) → R3
  70. If($K3 ≟ 1, O3) → S3
  71. If($K3 ≟ 1, P3) → T3
  72. If($K3 ≟ 2, M3) → U3
  73. If($K3 ≟ 2, N3) → V3
  74. If($K3 ≟ 2, O3) → W3
  75. If($K3 ≟ 2, P3) → X3
  76. If($K3 ≟ 3, M3) → Y3
  77. If($K3 ≟ 3, N3) → Z3
  78. If($K3 ≟ 3, O3) → AA3
  79. If($K3 ≟ 3, P3) → AB3
  80. If($K3 ≟ 4, M3) → AC3
  81. If($K3 ≟ 4, N3) → AD3
  82. If($K3 ≟ 4, O3) → AE3
  83. If($K3 ≟ 4, P3) → AF3
  84. If($K3 ≟ 5, M3) → AG3
  85. If($K3 ≟ 5, N3) → AH3
  86. If($K3 ≟ 5, O3) → AI3
  87. If($K3 ≟ 5, P3) → AJ3
  88. If(Angle[E4, H4, I4] < 180° ∧ m > 0, Polygon[E4, H4, I4]) → M4
  89. If(Angle[H4, G4, I4] < 180° ∧ m > 0, Polygon[H4, G4, I4]) → N4
  90. If(Angle[G4, F4, I4] < 180° ∧ m > 0, Polygon[G4, F4, I4]) → O4
  91. If(Angle[F4, E4, I4] < 180° ∧ m > 0, Polygon[F4, E4, I4]) → P4
  92. If($K4 ≟ 1, M4) → Q4
  93. If($K4 ≟ 1, N4) → R4
  94. If($K4 ≟ 1, O4) → S4
  95. If($K4 ≟ 1, P4) → T4
  96. If($K4 ≟ 2, M4) → U4
  97. If($K4 ≟ 2, N4) → V4
  98. If($K4 ≟ 2, O4) → W4
  99. If($K4 ≟ 2, P4) → X4
  100. If($K4 ≟ 3, M4) → Y4
  101. If($K4 ≟ 3, N4) → Z4
  102. If($K4 ≟ 3, O4) → AA4
  103. If($K4 ≟ 3, P4) → AB4
  104. If($K4 ≟ 4, M4) → AC4
  105. If($K4 ≟ 4, N4) → AD4
  106. If($K4 ≟ 4, O4) → AE4
  107. If($K4 ≟ 4, P4) → AF4
  108. If($K4 ≟ 5, M4) → AG4
  109. If($K4 ≟ 5, N4) → AH4
  110. If($K4 ≟ 5, O4) → AI4
  111. If($K4 ≟ 5, P4) → AJ4
  112. If(Angle[E5, H5, I5] < 180° ∧ m > 0, Polygon[E5, H5, I5]) → M5
  113. If(Angle[H5, G5, I5] < 180° ∧ m > 0, Polygon[H5, G5, I5]) → N5
  114. If(Angle[G5, F5, I5] < 180° ∧ m > 0, Polygon[G5, F5, I5]) → O5
  115. If(Angle[F5, E5, I5] < 180° ∧ m > 0, Polygon[F5, E5, I5]) → P5
  116. If($K5 ≟ 1, M5) → Q5
  117. If($K5 ≟ 1, N5) → R5
  118. If($K5 ≟ 1, O5) → S5
  119. If($K5 ≟ 1, P5) → T5
  120. If($K5 ≟ 2, M5) → U5
  121. If($K5 ≟ 2, N5) → V5
  122. If($K5 ≟ 2, O5) → W5
  123. If($K5 ≟ 2, P5) → X5
  124. If($K5 ≟ 3, M5) → Y5
  125. If($K5 ≟ 3, N5) → Z5
  126. If($K5 ≟ 3, O5) → AA5
  127. If($K5 ≟ 3, P5) → AB5
  128. If($K5 ≟ 4, M5) → AC5
  129. If($K5 ≟ 4, N5) → AD5
  130. If($K5 ≟ 4, O5) → AE5
  131. If($K5 ≟ 4, P5) → AF5
  132. If($K5 ≟ 5, M5) → AG5
  133. If($K5 ≟ 5, N5) → AH5
  134. If($K5 ≟ 5, O5) → AI5
  135. If($K5 ≟ 5, P5) → AJ5
  136. If(Angle[E6, H6, I6] < 180° ∧ m > 0, Polygon[E6, H6, I6]) → M6
  137. If(Angle[H6, G6, I6] < 180° ∧ m > 0, Polygon[H6, G6, I6]) → N6
  138. If(Angle[G6, F6, I6] < 180° ∧ m > 0, Polygon[G6, F6, I6]) → O6
  139. If(Angle[F6, E6, I6] < 180° ∧ m > 0, Polygon[F6, E6, I6]) → P6
  140. If($K6 ≟ 1, M6) → Q6
  141. If($K6 ≟ 1, N6) → R6
  142. If($K6 ≟ 1, O6) → S6
  143. If($K6 ≟ 1, P6) → T6
  144. If($K6 ≟ 2, M6) → U6
  145. If($K6 ≟ 2, N6) → V6
  146. If($K6 ≟ 2, O6) → W6
  147. If($K6 ≟ 2, P6) → X6
  148. If($K6 ≟ 3, M6) → Y6
  149. If($K6 ≟ 3, N6) → Z6
  150. If($K6 ≟ 3, O6) → AA6
  151. If($K6 ≟ 3, P6) → AB6
  152. If($K6 ≟ 4, M6) → AC6
  153. If($K6 ≟ 4, N6) → AD6
  154. If($K6 ≟ 4, O6) → AE6
  155. If($K6 ≟ 4, P6) → AF6
  156. If($K6 ≟ 5, M6) → AG6
  157. If($K6 ≟ 5, N6) → AH6
  158. If($K6 ≟ 5, O6) → AI6
  159. If($K6 ≟ 5, P6) → AJ6
  160. Midpoint(G6, E6) → B
  161. Midpoint(F4, H4) → D
  162. Midpoint(F5, H5) → E
  163. Midpoint(E3, G3) → F
  164. Midpoint(E2, G2) → G
  165. Midpoint(I2, G) → H
  166. Polygon(D7, D8, D5, D6) → poly1, d7, d8, d5, d6
  167. Polygon(D8, D4, D1, D5) → poly2, d8_1, d4, d1, d5_1
  168. Polygon(D7, D3, D2, D6) → poly3, d7_1, d3, d2, d6_1
  169. Polygon(D6, D5, D1, D2) → poly4, d6_2, d5_2, d1_1, d2_1
  170. Polygon(D7, D8, D4, D3) → poly5, d7_2, d8_2, d4_1, d3_1
  171. Polygon(D3, D4, D1, D2) → poly6, d3_2, d4_2, d1_2, d2_2
  172. Midpoint(O, J) → K
  173. Midpoint(S, R) → U
  174. Polygon(M, J, I) → poly10, i, m_1, j
  175. Polygon(M, O, I) → poly12, i_2, m_2, o_1
  176. Polygon(O, P, M) → poly13, m_3, o_2, p_1
  177. Polygon(P, J, M) → poly14, m_4, p_2, j_2
  178. Vector(V, W) → a
  179. Vector(W, V) → b
  180. Midpoint(V, W) → A_1
  181. Midpoint(B_1, C_1) → D_1
  182. Vector(B_1, C_1) → c
  183. Vector(C_1, B_1) → d
  184. Vector(F_1, E_1) → e
  185. Vector(E_1, F_1) → f
  186. Midpoint(F_1, E_1) → G_1
  187. Segment(M, K) → g
  188. Line(O, g) → b_1
  189. Vector(H_1, I_1) → w_1
  190. Vector(I_1, H_1) → z_1
  191. Midpoint(I_1, H_1) → M_1
  192. Polygon(I, J, P, O) → poly11, i_1, j_1, p, o
  193. Polygon(I, J, P, O) → poly15, i_3, j_3, p_3, o_3
  194. Polygon(J_1, K_1, P_1, L_1) → poly16, j_4, k_1, p_4, l_1
  195. Midpoint(K_1, L_1) → Q_1
  196. Polygon(J_1, L_1, P_1, K_1) → poly17, j_5, l_2, p_5, k_2
  197. Line(M, m_3) → k
  198. Line(K, k) → a_1
  199. Intersect(k, b_1) → R_1
  200. Intersect(a_1, b_1) → S_1
  201. Segment(R_1, M) → c_1
  202. Segment(S_1, K) → d_1
  203. Midpoint(R_1, S_1) → T_1
  204. Vector(S_1, R_1) → u_1
  205. Vector(R_1, S_1) → v_1
  206. RotateText("B", 330°) → texte11
  207. Midpoint(N, Q) → B_2
  208. Midpoint(R, S) → C_2
  209. Vector(B_2, C_2) → g_1
  210. Translate(W_1, g_1) → D_2
  211. Translate(Z_1, g_1) → E_2
  212. Translate(Q, g_1) → Q'
  213. Translate(N, g_1) → N'
  214. Polygon(D_2, E_2, Q', N') → poly18', d_2, e_2, q', n'
  215. Segment(E_2, Z_1) → t_1
  216. Segment(D_2, W_1) → a_2
  217. Midpoint(E_2, R) → N_1
  218. Midpoint(N_1, I_2) → O_1
  219. Polygon(R, S, O_1) → poly8, o_4, r_1, s_1
  220. Vector(O_1, N_1) → z
  221. Translate(R, Vector[(If[Animation < 0, 0, Animation] * z)]) → R'
  222. Translate(S, Vector[(If[Animation < 0, 0, Animation] * z)]) → S'
  223. Translate(O_1, Vector[(If[Animation < 0, 0, Animation] * z)]) → U_1
  224. Polygon(R', S', U_1) → poly8', u_2, r', s'
  225. Midpoint(R', S') → V_1
  226. Vector(U, V_1) → t
  227. Translate(D_2, t) → F_2
  228. Translate(E_2, t) → G_2
  229. Translate(Q', t) → Q''
  230. Translate(N', t) → N''
  231. Polygon(F_2, G_2, Q'', N'') → poly18'', f_2, g_2, q'', n''
  232. Polygon(O_1, N, Q) → poly9, q_2, o_5, n_3
  233. Vector(O_1, I_2) → e_1
  234. Translate(O_1, Vector[(If[Animation < 0, 0, Animation] * e_1)]) → H_2
  235. Translate(N, Vector[(If[Animation < 0, 0, Animation] * e_1)]) → N'_1
  236. Translate(Q, Vector[(If[Animation < 0, 0, Animation] * e_1)]) → Q'_1
  237. Polygon(H_2, N'_1, Q'_1) → poly9', q'_1, h_2, n'_1
  238. Midpoint(N'_1, Q'_1) → J_2
  239. Vector(B_2, J_2) → f_1
  240. Translate(W_1, f_1) → K_2
  241. Translate(Z_1, f_1) → L_2
  242. Translate(Q, f_1) → Q'_2
  243. Translate(N, f_1) → N'_2
  244. Polygon(K_2, L_2, Q'_2, N'_2) → poly18'_1, k_3, l_3, q'_2, n'_2
  245. Polygon(H_2, L_2, Q'_1) → poly19, q'_3, h_1, l_4
  246. Polygon(H_2, K_2, N'_1) → poly20, n'_3, h_3, k_4
  247. Polygon(H_2, K_2, L_2) → poly21, l_5, h_4, k_5
  248. Polygon(K_2, L_2, Q'_1, N'_1) → poly22, k_6, l_6, q'_4, n'_4
  249. Polygon(F_2, U_1, R') → poly23, r'_1, f_3, u_3
  250. Polygon(F_2, G_2, U_1) → poly24, u_4, f_4, g_3
  251. Polygon(G_2, S', U_1) → poly25, u_5, g_4, s'_1
  252. Polygon(F_2, G_2, U_1) → poly26, u_6, f_5, g_5
  253. Polygon(F_2, G_2, S', R') → poly27, f_6, g_6, s'_2, r'_2
  254. Midpoint(G_2, R') → N_2
  255. Midpoint(L_2, N'_1) → O_2
  256. Segment(N_2, U_1) → c_2
  257. Segment(O_2, H_2) → r_2
  258. Vector(I_1, H_1) → v_2
  259. Midpoint(h_2) → P_2
  260. Segment(Q_2, R_2) → s_2
  261. Midpoint(d_2) → S_2
  262. Midpoint(p_4) → T_2
  263. Midpoint(r_2) → U_2
  264. Segment(V_2, W_2) → t_2
  265. Segment(Z_2, A_3) → a_3
  266. Polygon(View3D, B_3, C_3, D_3) → q1, view3d, b_3, c_3, d_3

AI 绘图提示语参考

绘制一个金字塔，并展示其体积公式证明。首先绘制底面多边形和顶点，然后连接顶点和底面各顶点，形成多个三角形。接着通过调整动画参数，动态展示体积计算过程，最后通过旋转和移动文本，标注关键几何关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。