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Volume Investigation

Volume Investigation
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2020-01-03 16:41
VectorTrigonometryCoordinate GeometrySolid GeometryGeometric Construction
This example demonstrates the exploration of volume through 3D geometry and vector calculations. It involves core mathematical concepts such as volume computation, trigonometry, equation solving, 2D vector operations, and set theory. Key GeoGebra commands and objects used include Vector, Sequence, Segment, Translate, Line, Intersect, Polygon, Point, Midpoint, If, and Element, which are essential for constructing the geometric models and animations. Suitable for high school students and educators, this tool aids in deepening understanding of solid geometry and vector mathematics.

GeoGebra Commands & Parameters

  1. Vector(CC, Z) → w
  2. Vector(CC, Y) → v
  3. Vector(CC, X) → u
  4. Sequence(O + ((r * cos(i)) * u) + ((r * sin(i)) * v) - (h / 2 * w), i, 0, 6.28319, (2 * π / n)) → liste1
  5. Segment(F, E) → j
  6. Segment(G, E) → i
  7. Translate(G, Vector[((3 * d) * Vector[G, F])]) → F'
  8. Line(F', j) → k
  9. Intersect(k, i) → I
  10. Translate(D, Vector[((3 * d) * Vector[D, K])]) → E'
  11. Segment(I, F') → l
  12. Sequence(A + ((r * cos(i)) * u) + ((r * sin(i)) * v), i, 0, 6.28319, (2 * π / n)) → liste2
  13. Sequence(Polygon[Element[liste1, i], Element[liste1, i + 1], Element[liste2, i + 1], Element[liste2, i]], i, 1, Length[liste1]) → liste3
  14. Sequence(B + ((r * cos(i)) * u) + ((r * sin(i)) * v) - (h / 2 * w), i, 0, 6.28319, (2 * π / n)) → liste4
  15. Sequence(Polygon[Element[liste4, i], Element[liste4, i + 1], D], i, 1, Length[liste1]) → liste5
  16. Polygon(liste1) → poly1
  17. Translate(liste1, Vector[(d * Vector[J, A])]) → liste7
  18. Segment(D, B) → c
  19. Sequence(E' + ((l * cos(i)) * u) + ((l * sin(i)) * v), i, 0, 6.28319, (2 * π / n)) → liste9
  20. Sequence(Polygon[Element[liste1, i], Element[liste1, i + 1], Element[liste7, i + 1], Element[liste7, i]], i, 1, Length[liste1]) → liste8
  21. Segment(F, G) → g
  22. Polygon(liste4) → poly2
  23. Polygon(liste2) → poly3
  24. Sequence(Segment[Element[liste1, i], Element[liste1, i + 1]], i, 1, Length[liste1]) → liste11
  25. Point(liste1) → H
  26. Segment(A, J) → a
  27. Segment(D, K) → m
  28. Sequence(Segment[Element[liste4, i], Element[liste4, i + 1]], i, 1, Length[liste4]) → liste12
  29. Polygon(liste7) → poly4
  30. Vector(J, K) → z
  31. Translate(H, z) → H'
  32. Segment(K, H') → s
  33. Segment(J, H) → q
  34. Sequence(Polygon[Element[liste4, i], Element[liste4, i + 1], Element[liste9, i + 1], Element[liste9, i]], i, 1, Length[liste1]) → liste10
  35. Vector(P, K) → p
  36. Vector(N, J) → t
  37. Sequence(P + ((r / 10 * cos(i)) * u) + ((r / 10 * sin(i)) * v), i, 0, 6.28319, 0.15707963267948966) → liste13
  38. Polygon(liste13) → poly5
  39. Sequence(N + ((r / 10 * cos(i)) * u) + ((r / 10 * sin(i)) * v), i, 0, 6.28319, 0.15708) → liste14
  40. Polygon(liste14) → poly6
  41. Midpoint(P, N) → L
  42. Sequence(S + ((r / 10 * cos(i)) * u) + ((r / 10 * sin(i)) * w), i, 0, 6.28319, 0.15708) → liste17
  43. Polygon(liste17) → poly7
  44. Vector(S, P) → u_1
  45. Vector(S, N) → v_1
  46. Sequence(Translate[poly7, Vector[(i * u_1)]], i, 0, 1, 0.01) → liste18
  47. Sequence(Translate[poly7, Vector[(i * v_1)]], i, 0, 1, 0.01) → liste19
  48. If(y(B) > y(O), 1, -1) → aa
  49. Sequence(Translate[poly1, Vector[(i * Vector[J, A])]], i, 0.3333333333333333, 0.6666666666666666, 0.33333) → liste6
  50. Sequence(Translate[poly6, Vector[(i * p)]], i, -0.15, 1, 0.05) → liste16
  51. Sequence(Translate[poly5, Vector[(i * p)]], i, -0.175, 1, 0.05) → liste15
  52. Element(liste1, Length[liste1] / 2) → PP
  53. Translate(PP, Vector[(0.3333 * Vector[J, A])]) → PP'
  54. Translate(PP, Vector[(2 / 3 * Vector[J, A])]) → PP'_1
  55. Sequence(Segment[Element[liste2, i], Element[liste2, i + 1]], i, 1, Length[liste2]) → liste20
  56. Polygon(View3D, AA, BB, CC) → poly8, view3d, aa_1, bb, cc