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Conjecture for the Volume of a Pyramid (Copy)

Conjecture for the Volume of a Pyramid (Copy)
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2017-03-13 23:58
PolygonSolid GeometryIntegralTransformationSlider Animation
① This GeoGebra example dynamically constructs a pyramid and related geometric elements to visually demonstrate the relationship between pyramid volume, base area, and height, guiding users to conjecture the volume formula. ② Core mathematical concepts include the pyramid volume formula (V = 1/3 × base area × height), definite integrals in volume computation, arithmetic mean, and connections to triangles and spheres. ③ Construction highlights: from the summary, commands such as Sequence, Element, Point, Polygon, Segment, PointIn, and Translate are used, creating multiple points, polygons, segments, and auxiliary objects (z, w, v, a, O, P, α, β, sinα, cosα), with a slider (echelle) for scaling and translation to create a copy (cube'), plus text objects for formulas. ④ Suitable for high school or early college students in solid geometry, exploratory learning of volume formulas, and as an intuitive introduction to definite integral applications.

GeoGebra Commands & Parameters

  1. Sequence(Mat * Element[cube, k], k, 1, 8) → cube'
  2. Element(Element[AE, 3], 1) → droite
  3. Sequence((Element[cube', k, 1, 1], Element[cube', k, 2, 1]), k, 1, 8) → listepoints
  4. Point(listepoints) → A
  5. Point(listepoints) → E
  6. Point(listepoints) → C_1
  7. Point(listepoints) → D
  8. Polygon(A, E, C_1, D) → poly1, a_1, b, c_1, d
  9. Point(listepoints) → B
  10. Point(listepoints) → F
  11. Point(listepoints) → G
  12. Point(listepoints) → H
  13. Polygon(B, F, G, H) → poly2, e, f, g, h
  14. Point(listepoints) → C
  15. Polygon(F, C, A, B) → poly3, f_1, i, a_2, b_1
  16. Element(AB, 3, 1) → avant
  17. Polygon(C_1, G, H, E) → poly4, c_2, g_1, h_1, e_1
  18. Element(AC, 3, 1) → dessus
  19. Polygon(F, C, C_1, G) → poly5, f_2, c_3, c_4, g_2
  20. Polygon(A, B, H, E) → poly6, a_3, b_2, h_2, e_2
  21. Segment(E, A) → c
  22. Segment(A, C) → j
  23. Segment(A, B) → k
  24. Segment(C, C_1) → l
  25. Segment(C_1, E) → m
  26. Segment(H, E) → n
  27. Segment(A, B) → p
  28. Segment(B, H) → q
  29. Segment(H, E) → r
  30. Segment(H, G) → s
  31. Segment(B, F) → t
  32. Segment(G, C_1) → d_1
  33. Segment(F, G) → i_1
  34. Segment(F, C) → j_1
  35. PointIn(poly5) → J
  36. Translate(J, Vector[val1 * Vector[A, B]]) → pyr1_1
  37. Translate(A, Vector[val1 * Vector[A, B]]) → pyr1_2
  38. Translate(B, Vector[val1 * Vector[A, B]]) → pyr1_3
  39. Translate(H, Vector[val1 * Vector[A, B]]) → pyr1_4
  40. Translate(E, Vector[val1 * Vector[A, B]]) → pyr1_5
  41. Polygon(pyr1_1, pyr1_2, pyr1_3) → poly11, pyr1_6, pyr1_7, pyr1_8
  42. Polygon(pyr1_1, pyr1_3, pyr1_4) → poly12, pyr1_9, pyr1_{10}, pyr1_{11}
  43. Polygon(pyr1_5, pyr1_1, pyr1_2) → poly13, pyr1_{12}, pyr1_{13}, pyr1_{14}
  44. Polygon(pyr1_5, pyr1_4, pyr1_3, pyr1_2) → poly14, pyr1_{15}, pyr1_{16}, pyr1_{17}, pyr1_{18}
  45. Translate(J, Vector[val2 * Vector[A, E]]) → pyr2_1
  46. Translate(E, Vector[val2 * Vector[A, E]]) → pyr2_2
  47. Translate(H, Vector[val2 * Vector[A, E]]) → pyr2_3
  48. Translate(C_1, Vector[val2 * Vector[A, E]]) → pyr2_4
  49. Translate(G, Vector[val2 * Vector[A, E]]) → pyr2_5
  50. Polygon(pyr2_1, pyr2_2, pyr2_3) → poly15, pyr2_6, pyr2_7, pyr2_8
  51. Polygon(pyr2_1, pyr2_4, pyr2_5) → poly16, pyr2_9, pyr2_{10}, pyr2_{11}
  52. Polygon(pyr2_1, pyr2_3, pyr2_5) → poly17, pyr2_{12}, pyr2_{13}, pyr2_{14}
  53. Polygon(pyr2_4, pyr2_2, pyr2_3, pyr2_5) → poly18, pyr2_{15}, pyr2_{16}, pyr2_{17}, pyr2_{18}
  54. Translate(J, Vector[val3 * Vector[D, A]]) → pyr3_1
  55. Translate(F, Vector[val3 * Vector[D, A]]) → pyr3_2
  56. Translate(B, Vector[val3 * Vector[D, A]]) → pyr3_3
  57. Translate(G, Vector[val3 * Vector[D, A]]) → pyr3_4
  58. Translate(H, Vector[val3 * Vector[D, A]]) → pyr3_5
  59. Polygon(pyr3_1, pyr3_3, pyr3_2) → poly19, pyr3_6, pyr3_7, pyr3_8
  60. Polygon(pyr3_2, pyr3_4, pyr3_5, pyr3_3) → poly20, pyr3_9, pyr3_{10}, pyr3_{11}, pyr3_{12}
  61. Polygon(pyr3_1, pyr3_2, pyr3_4) → poly21, pyr3_{13}, pyr3_{14}, pyr3_{15}
  62. Polygon(pyr3_1, pyr3_5, pyr3_4) → poly22, pyr3_{16}, pyr3_{17}, pyr3_{18}