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Modifiable Solid: Regular Pentagon Cross Sections Parallel to yAxis

Modifiable Solid: Regular Pentagon Cross Sections Parallel to yAxis
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2019-05-10 07:07
PolygonSolid GeometryCalculus
In collections GeoGebra 3D & AR: PreCalc & Calculus Resources Cross Sectional Specials (Calculus)
This example demonstrates a modifiable solid where cross sections parallel to the yAxis are regular pentagons. Core math concepts include calculus-based cross-section analysis, polygon properties in geometry, and integral applications of functions. Construction highlights involve GeoGebra commands like Intersect, Segment, Midpoint, Line, Sphere, etc., to build cross-sections and solid shapes. Suitable for high school and above students learning solid geometry and calculus-based cross-section problems.

GeoGebra Commands & Parameters

  1. Intersect(lower, h, (-10, -2)) → C
  2. Intersect(upper, h, (-10, 6.159318853398537)) → B
  3. Segment(C, B) → q
  4. Intersect(upper, q, (-10, 6.159318853398537)) → G
  5. Midpoint(C, G) → T
  6. Line(T, T + (1, 0, 0)) → l_1
  7. Segment(H, I) → s_1
  8. Midpoint(s_1) → E
  9. Line(E, E + (1, 0, 0)) → r_1
  10. Segment(J, K) → q_1
  11. Midpoint(q_1) → F
  12. Line(F, F + (1, 0, 0)) → c
  13. Line(B, D) → i
  14. Sphere(B, d) → e
  15. Segment(C, B) → p
  16. Textfield(upper) → InputBox1
  17. Textfield(lower) → InputBox2
  18. Textfield(a) → InputBox3
  19. Textfield(b) → InputBox4
  20. Segment(C, G) → k_1
  21. Rotate(J, (2 * pi / 5), Z, c) → J'
  22. Rotate(J', (2 * pi / 5), Z, c) → J''
  23. Rotate(J'', (2 * pi / 5), Z, c) → J'''
  24. Polygon(K, J, J', J'', J''') → poly1, k, j, j', j'', j'''
  25. Rotate(H, (2 * pi / 5), A_1, r_1) → H'
  26. Rotate(H', (2 * pi / 5), A_1, r_1) → H''
  27. Rotate(H'', (2 * pi / 5), A_1, r_1) → H'''
  28. Polygon(H, H', H'', H''', I) → poly2, h_2, h', h'', h''', i_1
  29. Rotate(G, (2 * pi / 5), B_1, l_1) → G'
  30. Rotate(G', (2 * pi / 5), B_1, l_1) → G''
  31. Rotate(G'', (2 * pi / 5), B_1, l_1) → G'''
  32. Polygon(C, G, G', G'', G''') → poly3, c_1, g, g', g'', g'''
  33. Segment(K, J''') → f
  34. Segment(J''', J'') → l
  35. Segment(J'', J') → m
  36. Segment(J', J) → n
  37. Segment(H''', I) → r
  38. Segment(H''', H'') → s
  39. Segment(H'', H') → t
  40. Segment(H', H) → f_1
  41. Segment(C, G''') → g_1
  42. Segment(G''', G'') → j_1
  43. Segment(G'', G') → m_1
  44. Segment(G', G) → n_1
  45. CurveCartesian(t, (0.5 * (upper(t) + lower(t))), ((0.5 * (upper(t) - lower(t))) * (tan((3 * pi / 10)) + 1 / cos((3 * pi / 10)))), t, a, b) → u
  46. CurveCartesian(t, (0.5 * (upper(t) + lower(t))) - (sin((3 * pi / 10)) * (upper(t) - lower(t))), ((upper(t) - lower(t)) * sin((2 * pi / 5))), t, a, b) → o
  47. CurveCartesian(t, (0.5 * (upper(t) + lower(t))) + (sin((3 * pi / 10)) * (upper(t) - lower(t))), ((upper(t) - lower(t)) * sin((2 * pi / 5))), t, a, b) → v
  48. Surface(u, lower(u) - ((cos((2 * pi / 5)) * (upper(u) - lower(u))) * v), (((upper(u) - lower(u)) * sin((2 * pi / 5))) * v), u, a, b, v, 0, 1) → w
  49. Surface(u, upper(u) + ((cos((2 * pi / 5)) * (upper(u) - lower(u))) * v), (((upper(u) - lower(u)) * sin((2 * pi / 5))) * v), u, a, b, v, 0, 1) → a_1
  50. Surface(u, (0.5 * (upper(u) + lower(u))) - ((upper(u) - lower(u)) * sin((3 * pi / 10))) + ((sin((3 * pi / 10)) * (upper(u) - lower(u))) * v), ((upper(u) - lower(u)) * sin((2 * pi / 5))) + (((upper(u) - lower(u)) * cos((3 * pi / 10))) * v), u, a, b, v, 0, 1) → b_1
  51. Surface(u, (0.5 * (upper(u) + lower(u))) + ((sin((3 * pi / 10)) * (upper(u) - lower(u))) * v), ((0.5 * (upper(u) - lower(u))) * (tan((3 * pi / 10)) + 1 / cos((3 * pi / 10)))) - (((upper(u) - lower(u)) * cos((3 * pi / 10))) * v), u, a, b, v, 0, 1) → d_1

AI Drawing Prompt Reference

Create a modifiable solid where cross-sections parallel to the y-axis are regular pentagons. Start by defining the upper and lower boundaries, then calculate key points and connect them to form segments. Construct regular pentagons based on these segments. Rotate these pentagons around a central axis to create a continuous solid structure. Finally, combine all the rotated pentagons to display the complete shape of the solid.
The prompt above was compiled from the original drawing commands and is for reference only.