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Calculating the Surface Area of Prisms Using Nets

Calculating the Surface Area of Prisms Using Nets
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2023-08-10 04:47
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This example demonstrates how to unfold nets and calculate the surface area of triangular and rectangular prisms through a step-by-step exploration activity. The math concepts involved include nets, surface area calculation, and polygon area. Construction highlights derived from the GeoGebra command summary: commands like Shuffle, Zip, Mod, If, Polygon, Prism, Net, Translate, Line, Text, Append, OrthogonalLine, Segment, TableText, Vector; objects such as 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. Suitable for grades 6-8, used to learn how to find the surface area of prisms, pyramids, and cylinders using nets or formulas in numerical and word problems.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw the nets of a triangular prism and a square prism. First, draw a triangular prism with a triangular base, then draw a square prism with a square base. Next, unfold the nets of both prisms and label their bases, sides, and heights. Finally, ensure that each face in the nets is correctly connected to form complete nets.
The prompt above was compiled from the original drawing commands and is for reference only.