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Calculating Surface Area of Cylinders Using Nets

Calculating Surface Area of Cylinders Using Nets
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2022-11-16 00:58
Solid GeometryGeometric ConstructionSlider Animation
This example demonstrates how to calculate the surface area of cylinders by unfolding their nets. The key mathematical concepts include the geometry of cylinders, net unfolding, and surface area formulas. Construction involves using GeoGebra commands like Shuffle, Element, Rotate, and Circle to create and manipulate the cylinder model. It is suitable for students in grades 6-8 to understand solid geometry and surface area calculations.

GeoGebra Commands & Parameters

  1. Shuffle(Sequence[5]) → randomList2, {2, 4, 3, 5, 1}
  2. Element(randomList2, Mod[countQuestion, 5] + 1) → randomNum2
  3. Shuffle(Flatten[Append[Factors[randomNum2] \ {randomNum2}, 1]]) → l3, {1, 2, 2}
  4. Element(l3, 1) → Div
  5. Shuffle(Sequence[2, 5]) → randomList1, {3, 5, 2, 4}
  6. Element(randomList1, Mod[countQuestion, 4] + 1) → randomNum1
  7. Zip(Element[legends, kk], kk, clickedSurfaces) → legendsToShow
  8. Zip(Element[formulas, kk], kk, clickedSurfaces) → formulasToShow
  9. If(abs(tCil) < 1, ((rCil * sin((-θCil))), ((rCil * (1 - cos((-θCil)))) * cos(ϕCil)) - (hei * sin(ϕCil)), (hei * cos(ϕCil)) + ((rCil * (1 - cos((-θCil)))) * sin(ϕCil))), ((-pi), ((-hei) * sin(ϕCil)), (hei * cos(ϕCil)))) → CE
  10. If(abs(tCil) < 1, ((rCil * sin(θCil)), ((rCil * (1 - cos(θCil))) * cos(ϕCil)) - (hei * sin(ϕCil)), (hei * cos(ϕCil)) + ((rCil * (1 - cos(θCil))) * sin(ϕCil))), (pi, ((-hei) * sin(ϕCil)), (hei * cos(ϕCil)))) → CF
  11. If(abs(tCil) < 1, ((rCil * sin((-θCil))), ((rCil * (1 - cos((-θCil)))) * cos(ϕCil)), ((rCil * (1 - cos((-θCil)))) * sin(ϕCil))), ((-pi), 0, 0)) → CG
  12. If(abs(tCil) < 1, ((rCil * sin(θCil)), ((rCil * (1 - cos(θCil))) * cos(ϕCil)), ((rCil * (1 - cos(θCil))) * sin(ϕCil))), (pi, 0, 0)) → CH
  13. Segment(CE, CG) → mCil
  14. Segment(CF, CH) → nCil
  15. If(abs(tCil) < 1, CurveCartesian[(rCil * sin((CLIPBOARDmagicSTRINGu * θCil))), ((rCil * (1 - cos((CLIPBOARDmagicSTRINGu * θCil)))) * cos(ϕCil)), ((rCil * (1 - cos((CLIPBOARDmagicSTRINGu * θCil)))) * sin(ϕCil)), CLIPBOARDmagicSTRINGu, -1, 1], CurveCartesian[(pi * CLIPBOARDmagicSTRINGu), 0, 0, CLIPBOARDmagicSTRINGu, -1, 1]) → lCil
  16. If(abs(tCil) < 1, CurveCartesian[(rCil * sin((CLIPBOARDmagicSTRINGu * θCil))), ((rCil * (1 - cos((CLIPBOARDmagicSTRINGu * θCil)))) * cos(ϕCil)) - (hei * sin(ϕCil)), (hei * cos(ϕCil)) + ((rCil * (1 - cos((CLIPBOARDmagicSTRINGu * θCil)))) * sin(ϕCil)), CLIPBOARDmagicSTRINGu, -1, 1], CurveCartesian[(pi * CLIPBOARDmagicSTRINGu), ((-hei) * sin(ϕCil)), (hei * cos(ϕCil)), CLIPBOARDmagicSTRINGu, -1, 1]) → jCil
  17. If(abs(tCil) < 1, Surface[(rCil * sin((CLIPBOARDmagicSTRINGu * θCil))), ((rCil * (1 - cos((CLIPBOARDmagicSTRINGu * θCil)))) * cos(ϕCil)) - (CLIPBOARDmagicSTRINGv * sin(ϕCil)), (CLIPBOARDmagicSTRINGv * cos(ϕCil)) + ((rCil * (1 - cos((CLIPBOARDmagicSTRINGu * θCil)))) * sin(ϕCil)), CLIPBOARDmagicSTRINGu, -1, 1, CLIPBOARDmagicSTRINGv, 0, hei], Surface[(pi * CLIPBOARDmagicSTRINGu), ((-CLIPBOARDmagicSTRINGv) * sin(ϕCil)), (CLIPBOARDmagicSTRINGv * cos(ϕCil)), CLIPBOARDmagicSTRINGu, -1, 1, CLIPBOARDmagicSTRINGv, 0, hei]) → iCil
  18. Rotate(Rotate[eCil, ϕCil, Line[CC, xAxis]], ϕCil, xAxis) → topCil
  19. Circle((0, 1, 0), 1, xOyPlane) → baseCil
  20. Circle((0, 1, 0), 1, xOyPlane) → baseCil2
  21. Segment((0, 1, 0), (1, 1, 0)) → segRad
  22. TableText(formulas) → text1
  23. Polygon({((-pi), 0, 0), (pi, 0, 0), (pi, (-hei), 0), ((-pi), (-hei), 0)}) → unfoldedCilinder
  24. Translate(baseCil2, Vector[(-(0, 2 + hei))]) → baseCil2'
  25. Zip(Element[formulasR, kk], kk, clickedSurfaces) → formulasToShowR
  26. Append("\phantom{=}", Sequence["=", ii, 1, Length[formulasToShow] - 1]) → equalSigns
  27. TableText(formulasToShowR, "r") → completeTable
  28. Text("\textsf{Click all sections of the net}", (-3.1, 4.8), true, true, -1, 1) → onboardingSurfaces
  29. Vector(G, H) → onboardingSegmentArrow1
  30. PointIn(baseCil2) → I
  31. Vector(G, I) → onboardingSegmentsArrow2
  32. Text(SADyn, B, true, true, -1, -1) → text8
  33. Zip(If[kk ≟ 2, ((2 * input1) * input2), kk ≟ 1, 0, input1^(2)], kk, clickedSurfaces) → l1
  34. Corner(5) → L
  35. Vector(E, F) → onboardingSurfaceArrow
  36. TableText(legendsToShow, Sequence["\hspace{110}\vspace{10}", kk, 1, Length[formulasToShow]], formulasToShow, "vlcr") → tableLeft
  37. TableText({(("\color{" * colorBlue) * "}\rule{35mm}{0.5mm}"), "\hspace{20}", (("\color{" * colorBlue) * "}\rule{45mm}{0.5mm}")}, {(("\color{" * colorBlue) * "}\textsf{Surface Area =}"), "", formulaBelowTable}, "lcr") → belowTable
  38. Corner(5) → A
  39. Text("\textsf{Click one}", G + (-0.5, 1), true, true) → onboardingSegments

AI Drawing Prompt Reference

Create a 3D cylinder model and a corresponding unfolded net for surface-area calculation. Show the cylinder with its circular base and the curved lateral surface represented as a strip, then unfold that curved surface into the plane by mapping along the angular direction and the height direction, forming a rectangular-like net whose boundary is symmetric at the left and right ends. Include the base and side relationships in the drawing so the net clearly corresponds to the cylinder. Add on-screen guidance text and a table/annotation area indicating that users should click the sections of the net to reveal the associated surface-area formulas. The final composition should include both the 3D cylinder and its 2D net with interactive-style labels.
The prompt above was compiled from the original drawing commands and is for reference only.