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How Volumes Change With Changing Dimensions – Cone, Cylinder, Sphere

How Volumes Change With Changing Dimensions – Cone, Cylinder, Sphere
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2016-10-18 04:51
FunctionSolid GeometryTransformationSlider AnimationGraphing
① This GeoGebra example dynamically demonstrates how the volumes of cones, cylinders, and spheres change as their dimensions (such as radius and height) are varied. ② Core mathematical concepts include solid geometry volume formulas, multiplication, proportional reasoning, and the power/quadratic relationships between dimensions and volume, with optional connections to logarithmic functions and Poisson distribution for advanced exploration. ③ Construction highlights: using GeoGebra commands such as Segment, Polygon, Ellipse, OrthogonalLine, Circle, Intersect, Point, Line, and Midpoint to build the base shapes, and creating linked objects (l_1, w_1, r_1, V_1, etc.) to relate dimensions to volume, with sliders or dynamic points to control changes and update volume values in real time. ④ Suitable for high school to early college students, this example helps visualize the power-law dependence of volume on linear dimensions, and serves as a teaching tool for deriving volume formulas and modeling with functions.

GeoGebra Commands & Parameters

  1. Segment(A_1, B_1) → a
  2. Segment(B_1, C_1) → b
  3. Segment(C_1, D_1) → c
  4. Segment(D_1, A_1) → d
  5. Polygon(A_1, B_1, C_1, D_1) → poly1, a_1, b_1, c_1, d_1
  6. Polygon(A_2, B_2, C_2, D_2) → poly2, a_2, b_2, c_2, d_2
  7. Ellipse(E, E_1, B) → f
  8. Segment(E, E_1) → g
  9. OrthogonalLine(E, g) → h
  10. Circle(E, l_1) → k
  11. Intersect(k, h) → C, D
  12. OrthogonalLine(E_1, g) → i
  13. Circle(E_1, l_1) → p
  14. Intersect(p, i) → F, G
  15. Segment(E + (-0.05, 0), D + (-0.05, 0)) → j
  16. Point(f) → I
  17. Segment(E, E_1) → e
  18. Line(I, j) → m
  19. Circle(I, l_1) → q
  20. Intersect(q, m) → H, J
  21. Ellipse(D, G_1, J) → r
  22. Segment(J, I) → n
  23. Polygon(E, E_1, G_1, D) → poly3, e_1, e_2, g_1, d_3
  24. Ellipse(R_1, R_2, K) → s
  25. Segment(R_1, R_2) → t
  26. OrthogonalLine(R_1, t) → f_1
  27. OrthogonalLine(R_2, t) → h_1
  28. Circle(R_1, l_2) → k_1
  29. Circle(R_2, l_2) → p_1
  30. Intersect(k_1, f_1) → L, M
  31. Intersect(p_1, h_1) → N, O
  32. Segment(R_1 + (-0.055, 0), M + (-0.055, 0)) → i_1
  33. Point(s) → P
  34. Line(P, i_1) → m_1
  35. Circle(P, l_2) → q_1
  36. Intersect(q_1, m_1) → Q, R
  37. Segment(P, R) → n_1
  38. Ellipse(M, O, R) → s_1
  39. Polygon(R_1, R_2, O, M) → poly4, r_3, r_4, o, m_2
  40. Midpoint(e) → S
  41. Segment(S, E_1) → ra_1
  42. Segment(R_1, R_2) → across
  43. Midpoint(across) → T
  44. Segment(T, R_2) → ra_2
  45. Segment(E_1 + (0.05, 0), G + (0.05, 0)) → l
  46. Segment(R_2 + (0.055, 0), O + (0.055, 0)) → j_1

AI Drawing Prompt Reference

Create a dynamic visualization showing how the volumes of cones, cylinders, and spheres change with varying dimensions. Display different base shapes and heights, controlled by sliders or dynamic points, with real-time volume updates. Include polygons, ellipses, and circles, with vertices determined by intersections and perpendicular lines. Use distinct colors to differentiate parts, label key dimensions and volume formulas, clearly demonstrating the power-law dependence of volume on linear dimensions.
The prompt above was compiled from the original drawing commands and is for reference only.