Volume of a Truncated Pyramid: Geometric Construction and Integral Demonstration

GeoGebra Commands & Parameters

1. Point(Circle[A, AB]) → B
2. Segment(A, B) → d
3. Point(Circle[A, BC]) → D
4. Segment(A, D) → e
5. Point(Circle[A, AA']) → A'
6. Segment(A, A') → f
7. Line(B, e) → g
8. Line(D, d) → h_3
9. Intersect(g, h_3) → C
10. Line(B, f) → i
11. Line(A', i) → j
12. Line(A', d) → k
13. Intersect(i, k) → B'
14. Line(B', g) → l_1
15. Line(C, i) → m
16. Intersect(l_1, m) → C'
17. Line(A', l_1) → n_2
18. Line(C', k) → p
19. Intersect(n_2, p) → D'
20. Polygon(A, B, C, D) → poly1, a_1, b_1, e_1, c_1
21. Polygon(A', B', C', D') → poly2, d_1, f_1, g_1, h_1
22. Polygon(B', B, C, C') → poly3, f_2, b_2, e_2, g_2
23. Polygon(A', A, B, B') → poly4, d_2, a_2, b_3, f_3
24. Polygon(D, C, C', D') → poly5, c_2, e_3, g_3, h_2
25. Segment(B, D') → a
26. Segment(D', B) → b
27. Point(Circle[A_1, AB_1]) → B_1
28. Segment(A_1, B_1) → d_5
29. Point(Circle[A_1, BC_1]) → D_1
30. Segment(A_1, D_1) → e_7
31. Point(Circle[A_1, AA'_1]) → A'_1
32. Segment(A_1, A'_1) → f_7
33. Line(B_1, e_7) → g_7
34. Line(D_1, d_5) → h_6
35. Intersect(g_7, h_6) → C_1
36. Line(B_1, f_7) → i_1
37. Line(A'_1, i_1) → j_1
38. Line(A'_1, d_5) → k_1
39. Intersect(i_1, k_1) → B'_1
40. Line(B'_1, g_7) → l_2
41. Line(C_1, i_1) → m_1
42. Intersect(l_2, m_1) → C'_1
43. Line(A'_1, l_2) → n_1
44. Line(C'_1, k_1) → p_1
45. Intersect(n_1, p_1) → D'_1
46. Polygon(A_1, B_1, C_1, D_1) → poly1_1, a_3, b_4, e_6, c_3
47. Polygon(A'_1, B'_1, C'_1, D'_1) → poly2_1, d_3, f_4, g_4, h_4
48. Polygon(B'_1, B_1, C_1, C'_1) → poly3_1, f_5, b_5, e_4, g_5
49. Polygon(A'_1, A_1, B_1, B'_1) → poly4_1, d_4, a_4, b_6, f_6
50. Polygon(D_1, C_1, C'_1, D'_1) → poly5_1, c_4, e_5, g_6, h_5
51. Segment(B_1, D'_1) → a_5
52. Segment(D'_1, B_1) → b_7
53. Polygon(A_1, A, B, B_1) → poly6, a_6, a_7, b_8, b_9
54. Polygon(C_1, C, B, B_1) → poly7, c_5, c, b_{10}, b_{11}
55. Polygon(A, A_1, D_1, D) → poly8, a_8, a_9, d_6, d_7
56. Polygon(D_1, C_1, B_1, A_1) → poly9, d_8, c_6, b_{12}, a_{10}
57. Polygon(A_1, D_1, D'_1, A'_1) → poly10, a_{11}, d_9, d'_1, a'_1
58. Polygon(A'_1, A', A, A_1) → poly11, a'_2, a', a_{12}, a_{13}
59. Polygon(A'_1, D'_1, D', A') → poly12, a'_3, d'_2, d', a'_4
60. Polygon(B'_1, B', B, B_1) → poly13, b'_1, b', b_{13}, b_{14}
61. Polygon(C'_1, C', B', B'_1) → poly14, c'_1, c', b'_2, b'_3
62. Polygon(A', A'_1, B'_1, B') → poly15, a'_5, a'_6, b'_4, b'_5
63. Polygon(D'_1, C'_1, C', D') → poly16, d'_3, c'_2, c'_3, d'_4
64. Segment(E, F) → h
65. Segment(G, H) → l
66. Segment(G, E) → q