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使用展开图计算棱柱的表面积

使用展开图计算棱柱的表面积
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2023-08-10 04:47
多边形立体几何几何作图滑块动画函数图像
本范例通过逐步探索活动演示如何展开棱柱的展开图并计算三角棱柱和矩形棱柱的表面积。涉及的核心数学概念包括展开图、表面积计算、多边形面积等。作图要点——从下方『构造摘要』提炼用到了哪些 GeoGebra 命令/对象：Shuffle, Zip, Mod, If, Polygon, Prism, Net, Translate, Line, Text, Append, OrthogonalLine, Segment, TableText, Vector；对象：selectedSurfaces, Bases2, Bases1, heights, labels, countQuestion, Case, expectedSurfaces, fold, Base1, Base2, RectBase, height, prism0, A, B, C, D, face1, face3, face4, face2, faceABCD, edge1, edge2, edge3, edge4, edgeAB, edgeBC, edgeCD, edgeAD, net0, V1, V2, V3, V4, V8, V7, V9, V10。适合的学段或用途：6-8年级，用于学习如何使用展开图或公式计算棱柱、金字塔和圆柱的表面积。

GeoGebra 命令及参数

  1. Shuffle(Sequence[3, 4]) → Bases2, {4, 3}
  2. Shuffle(Sequence[3, 6]) → Bases1, {5, 4, 6, 3}
  3. Shuffle(Sequence[3, 5]) → heights, {5, 3, 4}
  4. Zip((("\textsf{" * TT) * "}"), TT, Flatten[{"", Zip[("Rectangle " * tt), tt, {"A", "B", "C", "D", "E", "F"}], {"Rectangle A", "Triangle B", "Rectangle C", "Rectangle D", "Triangle E"}}]) → labels
  5. Mod(countQuestion, 2) → Case
  6. If(countQuestion ≟ 0, 3, Element[Bases1, Mod[countQuestion, Length[Bases1]] + 1]) → Base1
  7. If(countQuestion ≟ 0, 3, Element[Bases2, Mod[countQuestion, Length[Bases2]] + 1]) → Base2
  8. Polygon({(((-Base1)) / 2, ((-Base2)) / 2), (Base1 / 2, ((-Base2)) / 2), (Base1 / 2, Base2 / 2), (((-Base1)) / 2, Base2 / 2)}) → RectBase
  9. If(countQuestion ≟ 0, 4, Min[Element[heights, Mod[countQuestion, Length[heights]] + 1], 8 - Base2]) → height
  10. Prism(RectBase, height) → prism0, A, edge1, edge2, edge3, edge4, edgeAB, edgeBC, edgeCD, edgeAD, B, C, D, face1, face3, face4, face2, faceABCD
  11. Net(prism0, fold) → net0, V1, V12, V13, V14, V6, V5, face01, face05, face06, face03, face04, V2, face02, Edge3, Edge10, Edge4, Edge9, Edge12, Edge2, Edge13, Edge15, Edge11, V3, Edge17, Edge19, Edge5, Edge18, Edge16, Edge8, Edge14, Edge7, Edge1, Edge6, V4, V8, V7, V9, V10, V11
  12. Translate(V11, Vector[(1, 0, 0)]) → V11'
  13. Polygon({(((-Base1)) / 2, ((-Base2)) / 2), (Base1 / 2, ((-Base2)) / 2), Element[{(Base1 / 2, Base2 / 2), (0, Base2 / 2), (If[Base1 ≟ 2, 0, -1], Base2 / 2)}, PosVertex]}) → TrianBase
  14. Prism(TrianBase, height) → prism1, E, edge7, edgeEF, edgeFG, edgeEG, F, G, face5, face7, face6, faceEFG, edge5, edge6
  15. Net(prism1, fold) → net1, W1, W9, facet05, facet01, facet02, facet03, facet04, edget1, edget3, edget6, edget10, W2, edget2, edget11, edget12, edget4, edget13, edget14, edget8, edget9, edget5, edget7, W3, W10, W8, W4, W5, W6, W7
  16. Translate(W6, Vector[W7, W6] / Distance[W7, W6]) → W6'_{1}
  17. Translate(W3, Vector[W7, W6] / Distance[W7, W6]) → W3'
  18. Translate(V3, Vector[(1, 0, 0)]) → V3'_{2}
  19. Translate(V3, Vector[(1 + (fold * height), 0, 0)]) → V3'_{1}
  20. Translate(V2, Vector[(1 + (fold * height), 0, 0)]) → V2'
  21. Zip(Element[resultsPrism, kk], kk, selectedSurfaces) → resultsPrismOrdered
  22. Zip(floor(nn), nn, resultsPrismOrdered) → resultsPrismOrderedFloor
  23. Shuffle(Sequence[1, 3]) → posVerts, {2, 3, 1}
  24. Line(W8, W10) → line2
  25. Text(edget3, Midpoint[edget3], true, true) → text8
  26. Text(edget6, Midpoint[edget6], true, true) → text9
  27. Zip((If[nn ≟ expectedSurfaces + 1 ∧ calculate ≟ expectedSurfaces + 1, "+\;", "\phantom{+\;}"] * Element[formulas, kk]), kk, selectedSurfaces, nn, Sequence[expectedSurfaces + 1]) → formulasToShow
  28. Zip((If[nn ≟ expectedSurfaces + 1 ∧ calculate ≟ expectedSurfaces + 1, "+\;", "\phantom{+\;}"] * Element[labels, kk]), kk, selectedSurfaces, nn, Sequence[expectedSurfaces + 1]) → labelsToShow
  29. Text(Base1, (-1, ((-Base2)) / 2, -0.5 + fold / 2), true, true) → labelBase1folded
  30. Append("", Sequence["=", ii, 1, expectedSurfaces]) → equalList
  31. Append("", Take[Zip[Text[nn], nn, resultsPrismOrdered], 2]) → resultsPrismToShow
  32. OrthogonalLine(W3, edget1, space) → heiLine1
  33. Segment(Intersect[OrthogonalLine[W3, line2, space], line2], W9) → heiSegment2
  34. Text(Base1, Midpoint[W1, W2], true, true) → text7
  35. Line(W1, W2) → line1
  36. Segment(Intersect[OrthogonalLine[W3, line1, space], line1], W3) → heiSegment1
  37. Text(height, Midpoint[Edge19] + (0, 1, 0), true, true) → labelheightfolded
  38. Text(height, Midpoint[W5, W3] + (0, 1.2, 0), true, true) → text10
  39. Text(Base2, (Base1 / 2, Base2 / 2, -0.5 + fold / 2), true, true) → labelBase2folded
  40. Translate(V6, Vector[(0, (-fold), 1 - fold)]) → V6'_{1}
  41. Translate(V5, Vector[(0, (-fold), 1 - fold)]) → V5'_{1}
  42. Translate(V14, Vector[((-fold), 0, 1 - fold)]) → V14'
  43. Translate(V13, Vector[((-fold), 0, 1 - fold)]) → V13'
  44. Translate(V3, Vector[(0, 1, 0)]) → V3'
  45. Translate(V10, Vector[(0, 1, 0)]) → V10'
  46. Zip(If[kk ≟ 0, "", Text[kk]], kk, resultsPrismOrderedFloor) → l1_{1}
  47. Zip(nn - floor(nn), nn, resultsPrismOrdered) → resultsPrismOrderedDec
  48. Zip(Split[Text[dd], {"."}], dd, resultsPrismOrderedDec) → resultsPrismOrderedDecText
  49. Zip(If[Length[LL] ≥ 2, ("." * Element[LL, 2]), ""], LL, resultsPrismOrderedDecText) → l2_{1}
  50. Zip(Length[tt], tt, l2_{1}) → l3
  51. Zip(tt + (If[kk ≟ 0, "\phantom{.00}", kk ≟ 2, qq + "\phantom{0}", qq + ""]), kk, l3, tt, l1_{1}, qq, l2_{1}) → l4
  52. TableText(labelsToShow, Sequence[(("\hspace{" * If[Case ≟ 1, 0, 20]) * "}"), ii, 1, Length[labelsToShow]], formulasToShow, Take[equalList, 1, Max[calculate, 1]], Take[l4, 1, Max[calculate, 1]], "vlcrrr") → text11
  53. Vector(Midpoint[V2', V3'_{1}], V3'_{1}) → l1
  54. Vector(Midpoint[V2', V3'_{1}], V2') → l2
  55. Translate(W7, Vector[W1, W7] / height) → W7'
  56. Translate(W6, Vector[W1, W7] / height) → W6'
  57. Segment(W7', W6') → edget8'
  58. Vector(Midpoint[W7', W6'], W7') → c1
  59. Vector(Midpoint[W7', W6'], W6') → c2
  60. Text(("c=" * Length[edget7]), PosLabelc, true, true) → labelc
  61. Translate(V1, Vector[(0, -1 - (fold * (Base2 + height)), 0)]) → V1'
  62. Translate(V2, Vector[(0, -1 - (fold * (Base2 + height)), 0)]) → V2'_{1}
  63. Segment(V1', V2'_{1}) → Edge3'
  64. Vector(Midpoint[V1', V2'_{1}], V1') → w1
  65. Vector(Midpoint[V2'_{1}, V1'], V2'_{1}) → w2
  66. Text(((If[Case ≟ 1, "b", "l"] * "=") * Base2), PointLabell, true, true) → labell
  67. Text(((If[Case ≟ 1, "a", "w"] * "=") * Base1), (0, 0), true, true) → labelw
  68. Vector(Corner[labelw, 1], Corner[labelw, 3]) → labelwsize
  69. Translate(Midpoint[V1', V2'_{1}], Vector[((-labelwsize)) / 2 - (0, 0, (1 - fold) / 2)]) → posLabelw
  70. Vector(Corner[labell, 1], Corner[labell, 3]) → labellsize
  71. Translate(Midpoint[V2', V3'_{1}], Vector[(0.3, y(labellsize) / 2, 0)]) → posLabell
  72. Text(labell, posLabell, true, true) → labellpos
  73. Text(("h=" * height), (0, 0), true, true, 0, 0) → labelh
  74. Vector(Corner[labelh, 1], Corner[labelh, 3]) → labelhsize
  75. If(Case ≟ 0, V3'_{2}, W3') → EndPointh1
  76. If(Case ≟ 0, V11', W6'_{1}) → EndPointh2
  77. Vector(Midpoint[EndPointh1, EndPointh2], EndPointh2) → h1
  78. Vector(Midpoint[EndPointh1, EndPointh2], EndPointh1) → h2
  79. Translate(Midpoint[EndPointh1, EndPointh2], Vector[(0, (y(labelhsize) * If[Case ≟ 0 ∧ fold ≟ 1, 1 / 2, 1]), 0)]) → PosLabelh
  80. Text(labelh, PosLabelh, true, true) → labelhPos
  81. Text(("\hspace{6}" * labelw), posLabelw, true, true) → labelwpos
  82. Vector(Corner[labelc, 1], Corner[labelc, 3]) → labelcsize
  83. Translate(Midpoint[W7', W6'], Vector[((-x(labelcsize)), y(labelcsize), 0)]) → poslabelc
  84. Text(labelc, poslabelc, true, true) → labelcpos
  85. Text(text3, (-4.5, Base2 / 2 + height + 4, 0), true, true) → onboardingSegments
  86. Text(text5, (-6, Base2 / 2 + height + 4.2, 0), true, true) → onboardingSurfaces
  87. TableText({(("\color{" * colorBlue_{1}) * "}\rule{45mm}{0.5mm}"), "", (((((If[Case ≟ 0, "\hspace{12}", ""] * "\color{") * colorBlue_{1}) * "}\rule{") * If[Case ≟ 0, 50, 65]) * "mm}{0.5mm}")}, {(("\color{" * colorBlue_{1}) * "}\textsf{Surface Area =}"), "", (Sum[resultsPrismOrdered] * "\text{ sq. units}")}, "lcr") → text1
  88. Vector((0, 9.5, 0), (0, 7.5, 0)) → u
  89. Text("B", Centroid[face04] - (0.55, -0.5), true, true) → labelRecB
  90. Text("A", Centroid[face03] - (0.55, -0.5), true, true) → labelRecA
  91. Text("C", Centroid[face01] - (0.55, -0.5), true, true) → labelRecC
  92. Text("D", Centroid[face06] - (0.55, -0.5), true, true) → labelRecD
  93. Text("E", Centroid[face05] - (0.55, -0.5), true, true) → labelRecE
  94. Text("F", Centroid[face02] - (0.55, -0.5), true, true) → labelRecF
  95. Text("A", Centroid[facet03] - (0.55, -0.5), true, true) → labelFaceA
  96. Text("B", Centroid[facet05] - (0.55, -0.5), true, true) → labelFaceB
  97. Text("C", Centroid[facet02] - (0.55, -0.5), true, true) → labelFaceC
  98. Text("D", Centroid[facet01] - (0.55, -0.5), true, true) → labelFaceD
  99. Text("E", Centroid[facet04] - (0.55, -0.5), true, true) → labelFaceF
  100. Text(text6, Corner[text11, 2] + (-0.7, 0.5), true, true, -1, -1) → text14
  101. Text(text12, Corner[text11, 1] + (0, 0.5), true, true, 1, -1) → text13

AI 绘图提示语参考

绘制一个三棱柱和一个四棱柱的展开图。首先绘制一个底面为三角形的三棱柱，然后绘制一个底面为正方形的四棱柱。接着，将两个棱柱的展开图分别展开，并标注它们的底面、侧面和高度。最后，确保展开图中的每个面都正确连接，形成完整的展开图。
以上提示语为根据原始绘图命令整理而来，仅供参考。