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利用表面积公式探究正棱柱的展开与折叠

利用表面积公式探究正棱柱的展开与折叠
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2023-08-12 02:00
多边形立体几何变换几何作图
本范例演示如何通过展开和折叠正棱柱的网格（Net），直观计算底面积、侧面积和表面积。核心数学概念包括正棱柱的结构、底面积公式、侧面积公式及表面积公式的推导。作图利用GeoGebra的Polygon、Prism、Net、Segment、Vertex、Corner等命令构建三维棱柱并生成其平面展开图，配合Textfield、If、Sequence等实现随机题目生成与答案验证。适合六至八年级学生练习表面积概念与公式应用，培养空间想象力和几何直观。

GeoGebra 命令及参数

  1. Sequence(Random[0, 99] / 100, A, 1, 200) → allRandomSeeds, {0.46, 0.45, 0.97, 0.61, 0.31, 0.11, 0.94, 0.91, 0.03, 0.28, 0.12, 0.3, 0.03, 0.5, 0.11, 0.3, 0.73, 0.41, 0.35, 0.95, 0.82, 0.48, 0.25, 0.02, 0.38, 0.44, 0.99, 0.41, 0.71, 0.29, 0.27, 0.23, 0.7, 0.29, 0.92, 0.58, 0.61, 0.63, 0.95, 0.33, 0.46, 0.11, 0.86, 0.26, 0.17, 0.36, 0.15, 0.81, 0.31, 0.15, 0.52, 0.27, 0.75, 0.85, 0.62, 0.13, 0.74, 0.52, 0.28, 0.97, 0.89, 0.64, 0.66, 0.12, 0.86, 0.26, 0.83, 0.31, 0.01, 0.83, 0.21, 0.55, 0.14, 0.9, 0.04, 0.67, 0.77, 0.31, 0.5, 0.6, 0.86, 0.87, 0.02, 0.29, 0.03, 0.73, 0.79, 0.54, 0.03, 0.16, 0.76, 0.6, 0.52, 0.48, 0.42, 0.07, 0.97, 0.43, 0.9, 0.7, 0.51, 0.08, 0.89, 0.8, 0.26, 0.98, 0.18, 0, 0.64, 0.27, 0.8, 0.36, 0.58, 0.66, 0.41, 0.72, 0.91, 0.67, 0.71, 0.21, 0.62, 0.26, 0.66, 0.85, 0.6, 0.82, 0.63, 0.89, 0.83, 0.76, 0.43, 0.98, 0.5, 0.69, 0.07, 0.42, 0.18, 0.24, 0.05, 0.06, 0.94, 0.73, 0.09, 0.89, 0.31, 0.3, 0.18, 0.28, 0.54, 0.13, 0.36, 0.42, 0.84, 0.36, 0.74, 0.76, 0.77, 0.33, 0.84, 0.81, 0.16, 0.68, 0.01, 0.68, 0.3, 0.47, 0.75, 0.23, 0.62, 0.1, 0.32, 0.12, 0.14, 0.05, 0.2, 0.5, 0.81, 0.28, 0.69, 0.93, 0.21, 0.6, 0.68, 0.36, 0.99, 0.08, 0.45, 0.53, 0.18, 0.91, 0.59, 0.33, 0.08, 0.45, 0.29, 0.73, 0.81, 0.53, 0.45, 0.47}
  2. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  3. If(countQuestion ≟ 0, 1, 1 + floor((currentRandomSeeds(1) * 4))) → randomNum1
  4. If(countQuestion ≟ 0, 3, Max[1 + floor(((currentRandomSeeds(2) * 3) * randomNum1)), (1 / 2 * randomNum1)]) → randomNum2
  5. Element({(2 * randomNum1^(2)) + ((4 * randomNum1) * randomNum2), (randomNum1 * triHeight) + ((3 * randomNum1) * randomNum2)}, Case) → expected
  6. If(Case ≟ 1, expected, (randomNum1 * exactTriHeight) + ((3 * randomNum1) * randomNum2)) → exactAnswer
  7. Polygon({(0, 0), (1, 0), (1, 1), (0, 1)}) → poly1
  8. Polygon({(0, 0), (1, 0), (cos(60°), sin(60°))}) → poly2
  9. Prism(poly2, graphHeight) → prism2, B, edge3, edgeBC, edgeCD, edgeBD, C, D, face1, face3, face2, faceBCD, edge1, edge2
  10. Prism(poly1, graphHeight) → prism1, I, edge8, edge9, edge10, edge11, edgeIJ, edgeJK, edgeKL, edgeIL, J, K, L, face8, face10, face11, face9, faceIJKL
  11. Textfield(input2) → InputBox2
  12. Polygon(Zip[PixelsToPoint[Corner[1], Corner[3], Corner[5], A], A, {(pixelsLeft, pixelsTop), (pixelsLeft, y(Corner[5]) - pixelsBottom), Corner[5] - (pixelsRight, pixelsBottom), (x(Corner[5]) - pixelsRight, pixelsTop)}]) → borderBox
  13. Corner(5) → layoutAppSize
  14. onboardingTextOffset(Corner[onboardingText, 1], Corner[onboardingText, 3], onboardingCase, Corner[1], Corner[3], Corner[5]) → OnboardingTextOffset
  15. Textfield(input1) → InputBox1
  16. Textfield(input3) → InputBox3
  17. Intersect(inputOK = inputOK, inputOK = Sum[inputValid] ≟ Length[inputValid] ∧ (¬(Defined[IndexOf["?", inputStrings]] ∨ Defined[IndexOf["", inputStrings]]))) → A
  18. Sequence("", A, 1, numInputBoxes) → initializeStrings
  19. Sequence(false, A, 1, numInputBoxes) → initializeValid
  20. Net(prism1, fold) → net1, M, V, W, Z, A_1, B_1, face12, face13, face14, face15, face16, N, face17, edge12, edge13, edge14, edge15, edge16, edge17, edge18, edge19, edge20, O, edge21, edge22, edge23, edge24, edge25, edge26, edge27, edge28, edge29, edge30, P, Q, R, S, T, U
  21. Net(prism2, fold) → net2, E, K_1, face4, face18, face19, face20, face21, edge4, edge31, edge32, edge33, C_1, edge34, edge35, edge36, edge37, edge38, edge39, edge40, edge41, edge42, edge43, D_1, E_1, F_1, G_1, H_1, I_1, J_1
  22. Vector((0.84 + (0.16 * fold); (0.32 * (1 - fold)) + (pi / 2 * fold); ((-pi)) / 2 + ((0.8 * pi / 2) * (1 - fold)))) → view
  23. Corner(5) → AppSizeGV1
  24. Prism(poly1, heiHint) → prismHint1, F, edge5, edge6, edge7, edge44, edgeFG, edgeGH, edge46, edge45, G, H, L_1, face5, face7, face22, face6, face23
  25. Prism(poly2, heiHint) → prismHint2, M_1, edge49, edge50, edge52, edge51, N_1, O_1, face25, face27, face26, face24, edge47, edge48
  26. Text(randomNum1, (1.35 - fold, 0.4 - (0.2 * (1 - fold)), -0.3 + fold), true, true) → labelBase
  27. Text(randomNum2, (1, ((((-graphHeight)) / 2 + 0.2) * fold) + ((1 - fold) * (-0.2)), ((graphHeight / 2 + 0.2) * (1 - fold))), true, true) → labelHei
  28. Segment(K_1, Midpoint[E_1, F_1]) → segHeightTri
  29. Text(triHeight, K_1 + (0.2, 0.5), true, true) → labelTriHeight
  30. Polygon({0.5}) → poly3
  31. Polygon({Midpoint[Midpoint[E_1, F_1], E_1], Midpoint[E_1, F_1], Translate[Midpoint[E_1, F_1], Vector[(0.25 * Vector[Midpoint[E_1, F_1], K_1] / Length[segHeightTri])]], Translate[Midpoint[E_1, Midpoint[E_1, F_1]], Vector[(0.25 * Vector[Midpoint[E_1, F_1], K_1] / Length[segHeightTri])]]}) → q1
  32. Vertex(q1, 4) → P_1

AI 绘图提示语参考

绘制一个可展开和折叠的正棱柱展开图，包含一个正方形底面的四棱柱和一个正三角形底面的三棱柱，两者高度相同。展开时，棱柱的侧面和底面平铺在二维平面上，形成由多边形组成的平面网络；折叠时，恢复为立体棱柱。在展开图中，标注底面边长和棱柱高度的数值，并在三棱柱的展开图上用线段标出三角形底面的高。同时，在图形旁边显示两个输入框，用于输入计算出的底面积和侧面积，并有一个提示框显示随机生成的题目参数。整体布局包含一个用于交互的边框区域。
以上提示语为根据原始绘图命令整理而来，仅供参考。