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Cone and Cylinder Cross Sections

Cone and Cylinder Cross Sections
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2012-01-09 18:11
TrigonometrySolid GeometryGeometric Construction
This example demonstrates the cross-sectional shapes formed when a cone or cylinder is intersected by a plane parallel to its base. Key math concepts include solid geometry, cross-sections, similar figures, and properties of cones and cylinders. Construction highlights involve commands like OrthogonalLine, Point, Circle, Vector, Dilate, Conic, Translate, Polygon, Sequence, and more, suitable for high school students learning about solid geometry to visualize the formation and properties of cross-sectional figures.

GeoGebra Commands & Parameters

  1. OrthogonalLine(E, xAxis) → l
  2. Point(l) → P'
  3. OrthogonalLine(E, yAxis) → k
  4. Circle(O, Zoom) → c
  5. Point(k) → P
  6. Vector(O, Z) → w
  7. Vector(O, Y) → v
  8. Vector(O, X) → u
  9. Dilate(D, 0.6, O') → D'_3
  10. Dilate(C, 0.6, O') → C'_3
  11. Dilate(B, 0.6, O') → B'_3
  12. Dilate(A, 0.6, O') → A'_3
  13. Conic(Y, M_1, X, M_v, M_u) → d
  14. Conic(Z, X, M_w, M_2, M_u) → e
  15. Conic(Z, M_3, Y, M_w, M_v) → f
  16. Translate(A, Vector[(hauteur * u)]) → A'
  17. Translate(B, Vector[(hauteur * u)]) → B'
  18. Translate(C, Vector[(hauteur * u)]) → C'
  19. Translate(D, Vector[(hauteur * u)]) → D'
  20. Polygon(A', B', C', D') → poly1', a', b', c', d'
  21. Translate(B, Vector[(((1 - hauteur3) * hauteur) * u)]) → B'_2
  22. Translate(C, Vector[(((1 - hauteur3) * hauteur) * u)]) → C'_2
  23. Translate(D, Vector[(((1 - hauteur3) * hauteur) * u)]) → D'_2
  24. Translate(A, Vector[(((1 - hauteur3) * hauteur) * u)]) → A'_2
  25. Polygon(A'_2, B'_2, C'_2, D'_2) → poly1'', a'_2, b'_2, c'_2, d'_2
  26. Sequence(O' + ((b * v) * cos((360 / Nombre * k°) + i°)) + ((w * b) * sin((360 / Nombre * k°) + i°)), k, 0, Nombre) → liste1
  27. Translate(liste1, Vector[(hauteur * u)]) → liste2
  28. PointIn(poly1') → S
  29. Sequence(Segment[S, Element[liste1, k]], k, 0, Nombre) → liste3
  30. Sequence(Segment[Element[liste2, k], Element[liste1, k]], k, 0, Nombre) → liste4
  31. Element(liste1, 1) → P_1
  32. Element(liste1, 2) → P_2
  33. Element(liste1, 3) → P_3
  34. Element(liste1, 4) → P_4
  35. Element(liste1, 5) → P_5
  36. Conic(P_1, P_2, P_3, P_4, P_5) → h
  37. Dilate(h, hauteur3, S) → h'_1
  38. Translate(S, Vector[((-hauteur) * u)]) → S'
  39. Segment(S', S) → j
  40. Translate(A, Vector[((1 - hauteur3) * u)]) → A'_1
  41. Translate(B, Vector[((1 - hauteur3) * u)]) → B'_1
  42. Translate(C, Vector[((1 - hauteur3) * u)]) → C'_1
  43. Translate(D, Vector[((1 - hauteur3) * u)]) → D'_1
  44. Translate(S, Vector[(((-hauteur) * hauteur3) * u)]) → S''
  45. Translate(h, Vector[(hauteur * u)]) → h'
  46. Translate(h, Vector[(((1 - hauteur3) * hauteur) * u)]) → i_1
  47. Sequence(Dilate[h, l / n, S], l, 0, n) → cercles
  48. Polygon(B, C, D, A) → poly1, b_1, c_1, d_1, a
  49. Sequence(Translate[h, Vector[((l * hauteur / n) * u)]], l, 0, n) → cercle2

AI Drawing Prompt Reference

Draw a 3D solid geometry diagram showing a cone and a cylinder with parallel circular bases. The cone's apex is above the cylinder's base, and the cylinder consists of two parallel circular faces connected by vertical side lines. Show internal cross-sections: a plane parallel to the base cuts the cone, producing a smaller circle, and another parallel plane cuts the cylinder, producing a circle equal to the base. Draw radial lines from the cone's apex to the base edge to represent slant heights. Use dashed or semi-transparent lines for hidden parts. Highlight the similarity between the cone's cross-section and the base, and indicate height proportions.
The prompt above was compiled from the original drawing commands and is for reference only.