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Proof without words: volume of a pyramid

Proof without words: volume of a pyramid
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2019-12-27 05:09
VectorCoordinate GeometrySolid GeometryGeometric Construction
This example dynamically demonstrates the proof of the volume formula for a pyramid, involving core concepts like volume, geometric solids, and proportional relationships. It utilizes GeoGebra commands such as Midpoint, Polygon, Line, Vector, If, Sort, Segment, Intersect, RotateText, and Translate. Suitable for high school students and math enthusiasts studying solid geometry volume calculations.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a pyramid and demonstrate its volume formula proof. Start by drawing the base polygon and the apex, then connect the apex to each vertex of the base to form several triangles. Use the animation parameter to dynamically show the volume calculation process, and finally rotate and move text to label key geometric relationships.
The prompt above was compiled from the original drawing commands and is for reference only.