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Unfolding Cylinders

Unfolding Cylinders
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2022-11-11 22:37
Solid GeometryGeometric ConstructionSlider Animation
This example demonstrates how to unfold the lateral surface of a cylinder and create a net. By adjusting the dimensions of the cylinder, one can explore its geometric properties. The key mathematical concepts include the development of cylinders, surface area calculations, and geometric construction. The construction involves using GeoGebra commands such as Point, Circle, Ray, Polygon, Net, and Cylinder to build the cylinder model. It is suitable for middle school mathematics teaching, particularly for grades 6 to 8, to help students understand solid geometry and geometric applications.

GeoGebra Commands & Parameters

  1. Point(yAxis) → CA
  2. Point(yAxis) → CB
  3. Circle(CA, CB) → cCil
  4. Point(cCil) → C
  5. Ray(A, (x(A), y(A), 40)) → h
  6. Point(yAxis) → B
  7. Sequence(Rotate[C, ((t * 2) * pi / n), A], t, 1, n) → vert
  8. Polygon(vert) → poly
  9. OrthogonalLine(C, xOyPlane) → f
  10. Segment(HC, (x(C), y(C), 4)) → g
  11. Point(g) → E
  12. Prism(poly, Segment[C, E]) → a, F, face3, face4, face5, face6, face2, face13, face15, faceFGHIJK, edge1, edge2, G, edge3, edge4, edge5, edge6, edge13, edge16, edgeFG, edgeGH, edgeHI, edgeIJ, H, edgeJK, edgeFK, edge14, edge17, I, J, K, L, M, face1
  13. Point(Segment[(0, 0, 0.1), (0, 0, 4)]) → CD
  14. Pyramid(poly, CD) → b, face7, edge8, edge9, edge10, edge11, edge12, edge15, edge18, face9, face10, face11, face12, face8, face14, face16, edge7
  15. Net(a, (-open)) → prismNet, N, W, Z, A_1, B_1, C_1, D_1, E_1, F_1, G_1, H_1, O, I_1, J_1, K_1, L_1, M_1, N_1, O_1, P_1, Q_1, R_1, P, S_1, face17, face18, face19, face20, face21, face22, face23, face24, face34, Q, face35, edge19, edge20, edge21, edge22, edge23, edge24, edge25, edge26, edge27, R, edge28, edge29, edge30, edge31, edge32, edge33, edge34, edge35, edge36, edge37, S, edge38, edge39, edge40, edge41, edge42, edge43, edge44, edge45, edge70, edge71, T, edge72, edge73, edge74, edge75, edge76, edge77, edge78, edge79, edge80, edge81, U, V
  16. Net(b, (-open)) → pyrNet, T_1, E_2, F_2, G_2, H_2, I_2, J_2, K_2, face25, face26, face27, U_1, face28, face29, face30, face31, face32, face33, edge46, edge47, edge48, edge49, V_1, edge50, edge51, edge52, edge53, edge54, edge55, edge56, edge57, edge58, edge59, W_1, edge60, edge61, edge62, edge63, edge64, edge65, edge66, edge67, edge68, edge69, Z_1, A_2, B_2, C_2, D_2
  17. Random(3, 8) → nRand
  18. Circle(CA, CB) → d
  19. Cylinder(cCil, 3) → bCil, eCil, fCil
  20. Line(CC, xAxis) → aCil
  21. OrthogonalLine(CD, xOyPlane) → gCil
  22. Circle(gCil, CC) → kCil
  23. Rotate(Rotate[eCil, ϕCil, aCil], ϕCil, xAxis) → eeCil
  24. Rotate(Rotate[eCil, ϕCil, aCil], ϕCil, xAxis) → dCil
  25. Circle(CA, CB) → hCil
  26. If(abs(tCil) < 1, ((rCil * sin((-θCil))), ((rCil * (1 - cos((-θCil)))) * cos(ϕCil)) - (3 * sin(ϕCil)), (3 * cos(ϕCil)) + ((rCil * (1 - cos((-θCil)))) * sin(ϕCil))), ((-pi), (-3 * sin(ϕCil)), (3 * cos(ϕCil)))) → CE
  27. If(abs(tCil) < 1, ((rCil * sin(θCil)), ((rCil * (1 - cos(θCil))) * cos(ϕCil)) - (3 * sin(ϕCil)), (3 * cos(ϕCil)) + ((rCil * (1 - cos(θCil))) * sin(ϕCil))), (pi, (-3 * sin(ϕCil)), (3 * cos(ϕCil)))) → CF
  28. If(abs(tCil) < 1, ((rCil * sin((-θCil))), ((rCil * (1 - cos((-θCil)))) * cos(ϕCil)), ((rCil * (1 - cos((-θCil)))) * sin(ϕCil))), ((-pi), 0, 0)) → CG
  29. If(abs(tCil) < 1, ((rCil * sin(θCil)), ((rCil * (1 - cos(θCil))) * cos(ϕCil)), ((rCil * (1 - cos(θCil))) * sin(ϕCil))), (pi, 0, 0)) → CH
  30. Segment(CE, CG) → mCil
  31. Segment(CF, CH) → nCil
  32. Line((0, 1, 0), zAxis) → pCil
  33. Circle(pCil, CB) → qCil
  34. Circle(pCil, CB) → sCil
  35. OrthogonalLine((1, 0, 0), xOyPlane) → aCon
  36. Circle(aCon, (0, 0, 0)) → cCon
  37. Point(aCon) → KA
  38. Intersect(xAxis, aCon) → KB
  39. Cone(KB, KA, 1) → bCon, dCon, eCon
  40. Intersect(yAxis, xAxis) → KC
  41. Segment(KC, KA) → lCon
  42. Distance(KB, KC) → rCon
  43. Angle(KA, KC, KB) → α
  44. Rotate(KA, (abs(tCil) * α), yAxis) → KAA
  45. OrthogonalLine(KAA, xOyPlane) → fCon
  46. Intersect(fCon, xOyPlane) → KBB
  47. Rotate(KC, bet / 2, fCon) → KCC
  48. Rotate(KC, ((-bet)) / 2, fCon) → KCR
  49. CircumcircleArc(KCC, KC, KCR) → gCon
  50. Distance(KAA, KBB) → hCon
  51. Distance(KBB, KCC) → rcCon
  52. Surface(((kCon - ((v * rcCon) * cos((u * bet / 2)))) * cos(ϕCon)) - (((1 - v) * hCon) * sin(ϕCon)), ((v * rcCon) * sin((u * bet / 2))), (((1 - v) * hCon) * cos(ϕCon)) + ((kCon - ((v * rcCon) * cos((u * bet / 2)))) * sin(ϕCon)), u, -1, 1, v, 0, 1) → jCon
  53. CurveCartesian(((kCon - (rcCon * cos((u * bet / 2)))) * cos(ϕCon)), (rcCon * sin((u * bet / 2))), ((kCon - (rcCon * cos((u * bet / 2)))) * sin(ϕCon)), u, -1, 1) → mCon
  54. Segment(KE, KF) → nCon
  55. Segment(KD, KF) → pCon
  56. If(abs(tCil) < 1, CurveCartesian[(rCil * sin((u * θCil))), ((rCil * (1 - cos((u * θCil)))) * cos(ϕCil)) - (3 * sin(ϕCil)), (3 * cos(ϕCil)) + ((rCil * (1 - cos((u * θCil)))) * sin(ϕCil)), u, -1, 1], CurveCartesian[(pi * u), (-3 * sin(ϕCil)), (3 * cos(ϕCil)), u, -1, 1]) → jCil
  57. If(abs(tCil) < 1, CurveCartesian[(rCil * sin((u * θCil))), ((rCil * (1 - cos((u * θCil)))) * cos(ϕCil)), ((rCil * (1 - cos((u * θCil)))) * sin(ϕCil)), u, -1, 1], CurveCartesian[(pi * u), 0, 0, u, -1, 1]) → lCil
  58. If(abs(tCil) < 1, Surface[(rCil * sin((u * θCil))), ((rCil * (1 - cos((u * θCil)))) * cos(ϕCil)) - (v * sin(ϕCil)), (v * cos(ϕCil)) + ((rCil * (1 - cos((u * θCil)))) * sin(ϕCil)), u, -1, 1, v, 0, 3], Surface[(pi * u), ((-v) * sin(ϕCil)), (v * cos(ϕCil)), u, -1, 1, v, 0, 3]) → iCil

AI Drawing Prompt Reference

To draw an unfolded net of a cylinder, showing the lateral surface as a rectangle and two circles as the bases. Adjusting the cylinder dimensions allows exploration of geometric properties such as surface area. The construction process involves first creating the cylinder model, then unfolding the lateral surface to generate the net. The net includes a rectangle and two circles, clearly illustrating the three-dimensional structure of the cylinder. This is suitable for middle school mathematics teaching to help students understand solid geometry and geometric applications.
The prompt above was compiled from the original drawing commands and is for reference only.