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用金字塔表面积公式：展开与折叠网图计算底面积、侧面积与总表面积

用金字塔表面积公式：展开与折叠网图计算底面积、侧面积与总表面积
几何画板及绘图内容加载中，请稍等……
2024-03-22 02:26
多边形向量立体几何几何作图
该范例通过“展开—折叠”的方式展示正金字塔（right pyramid）的网图如何对应到表面积计算。核心数学概念包括：金字塔底面面积、侧面（若为直角/正多边形底则为三角形侧面）面积、以及由底面积与侧面积相加得到的总表面积；同时体现了几何不变量与网图对应关系。作图要点：利用随机参数生成几何尺寸（RandomUniform、Textfield），创建底面多边形与侧面三角形（Polygon、Pyramid、Segment），通过对顶点与中点等构造点进行度量与定位（Centroid、Midpoint、Distance），再将网图进行空间/平面展开与调整（Net、Translate、Rotate、Vector）；用条件分支控制保留/更新对象（If、KeepIf），并用标注文本呈现计算所需信息（Text、TextToUnicode）。适合用于中学阶段立体几何教学（约 GR 6-8），也可作为课堂演示、作业训练或公式应用探究活动。

GeoGebra 命令及参数

  1. RandomUniform(0, 0.99, 200) → allRandomSeeds, {0.3958756055353063, 0.8533786357676533, 0.004044199219652898, 0.2739489007846822, 0.32390929955138464, 0.6400266049391629, 0.7144932453044555, 0.8056432312578217, 0.2836550889873928, 0.23858909772382403, 0.28324361446946433, 0.05079331846618933, 0.4993192527327668, 0.49420030293256073, 0.9600861842660023, 0.0051241733203633765, 0.521083186262596, 0.2404841840370241, 0.03267929771955199, 0.13745159194388704, 0.9328227445246845, 0.23979697671322595, 0.9657137852929928, 0.03288983455700257, 0.12311827833552909, 0.9629842381399758, 0.2565096376804113, 0.6455830752708474, 0.9083147508516813, 0.2926144708655247, 0.03603258468383078, 0.19652024598418255, 0.32177210780988463, 0.8736420249282592, 0.9682948830339475, 0.8043254256503664, 0.49351495290328723, 0.2876288835017463, 0.5047653538012109, 0.5164382523662225, 0.7492456743589369, 0.08102090679579474, 0.6337643307619925, 0.9148076850494951, 0.9271291801470278, 0.27452665317016933, 0.08785596719447065, 0.05411852202088994, 0.10488787842817364, 0.7293867235093806, 0.6049744064914199, 0.7616547675402011, 0.5585640800634707, 0.7572498502853541, 0.2514335842239421, 0.7561747606547239, 0.08781271614431484, 0.3534871421762426, 0.7670688242780163, 0.03863218592880263, 0.590780804639442, 0.019077821588354416, 0.6045464335723592, 0.22022389087201888, 0.7345108763194677, 0.023691627656309364, 0.5863515153662204, 0.6530789255369284, 0.4743737138736718, 0.2922687823970698, 0.0641266296319264, 0.5959250480561464, 0.2982920366916876, 0.8492656005191535, 0.19588343221006665, 0.1312775716094131, 0.6684598789549232, 0.9187849550949932, 0.04714664207547015, 0.6920925599512389, 0.18489318658938406, 0.27505217630777856, 0.362313998429265, 0.24440945806696082, 0.04146559927303947, 0.4561242069000696, 0.24242851567691348, 0.9782806500017874, 0.8560895608277569, 0.8443348983116223, 0.425158318361831, 0.7722437006685167, 0.18239554627323812, 0.22978238282397684, 0.5320668378693099, 0.9267388621284618, 0.6006437554351702, 0.7881549682520774, 0.4772309537675705, 0.049208509267538246, 0.430670711433712, 0.10136877791438538, 0.10992602740436652, 0.06813624941676814, 0.409428879786247, 0.39885651247617043, 0.44244731495915984, 0.4249929562521509, 0.35012107379593876, 0.23889923235742483, 0.932173509559451, 0.5219529924734885, 0.6190621492360409, 0.9731328791125432, 0.8347034960433505, 0.18932806054608944, 0.8802187881479377, 0.1532894926869308, 0.11230897507619898, 0.0999378366030312, 0.97588420487431, 0.0873480274866843, 0.3290454761557678, 0.28809509134564354, 0.595137816665954, 0.3725726071984496, 0.6706513399092382, 0.9685296963874221, 0.17962070463967056, 0.08540596836918167, 0.37383886304734765, 0.9454101206192298, 0.5787263288161181, 0.49286395032821834, 0.0057438906360658916, 0.6936419717196365, 0.12356170107097328, 0.37694191066983124, 0.5901497542303661, 0.1838327506730329, 0.980977194859502, 0.46259681059864877, 0.8426386646488337, 0.05870794600222147, 0.49559495900371675, 0.8211085230919339, 0.6320479436746284, 0.5423478171510185, 0.28908300840324475, 0.324052572078945, 0.5605211769454356, 0.9143858031848416, 0.3419291784294029, 0.15873059017980212, 0.6690433074243902, 0.8392681929267805, 0.6736614132817266, 0.8137084755949057, 0.18963625497373532, 0.2358214647029713, 0.6728427521243387, 0.43310186220577246, 0.8193662494510643, 0.5647074711273012, 0.2482365137398107, 0.5404266429610265, 0.36497580567941323, 0.6708681296668827, 0.45729444370205174, 0.8671971349192765, 0.1103032550184504, 0.1847706748743121, 0.8475625783905448, 0.5580076971444696, 0.28168979991257825, 0.6901603136790426, 0.3807315788154499, 0.5525732184588454, 0.7392983959951726, 0.6871116023336113, 0.7778758095432655, 0.666132418223546, 0.10468117295035269, 0.9574414540309326, 0.27793183274382793, 0.8427322076294166, 0.8870317542542345, 0.0050050652991491425, 0.6496601154597305, 0.7312149967085962, 0.6023455206064645, 0.4260432085807737, 0.1602145390142153, 0.5712204399310481, 0.1711858657569975, 0.04458281430410831, 0.1610129005446904, 0.7661910954295995, 0.9519167248734645, 0.4649147858317669}
  2. Polygon({(-0.5, 1, 0), (0.5, 1, 0), (0.5, 0, 0), (-0.5, 0, 0)}) → baseSqare
  3. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  4. If(currentRandomSeeds(3) ≤ 0.5 ∨ countQuestion ≟ 0, 4, 3) → Case
  5. If(countQuestion ≟ 0, 4, floor((currentRandomSeeds(2) * 5)) + 1) → randomSideLength
  6. KeepIf(Mod[randomSideLength, kk] ≟ 0, kk, Sequence[randomSideLength]) → l1
  7. If(countQuestion ≟ 0, 1, Element[l1, floor((currentRandomSeeds(4) * Length[l1])) + 1]) → heiFact
  8. If(countQuestion ≟ 0, 2, (floor((currentRandomSeeds(1) * 3)) + 1) / heiFact) → hei_{1}
  9. Pyramid(baseSqare, (-hei_{1})) → b, T, face14, face16, face17, face15, edge26, edge27, edge28, edge29
  10. Net(b, tCil2) → d, U, face18, face20, face21, face19, edgeUV, edge33, edge36, edge30, edge31, edge34, V, edge35, edge37, edge38, edge39, edge40, edge32, L_1, S_1, T_1, U_1, V_1, W_1, face22
  11. Segment(Midpoint[edge36], V_1) → segTriHeight4
  12. Polygon((0, 0), (1, 0), 3) → poly1
  13. Centroid(poly1) → B
  14. Translate(poly1, Vector[B, (0, 1)]) → baseTri
  15. Pyramid(baseTri, hei_{1}) → a, C, face1, face3, face2, edge3, edge5, edge6
  16. Net(a, tCil2) → e, D, face4, edgeDE, edgeEF, edgeDF, edgeDW, edgeEW, edgeEZ, edgeFZ, edge11, edge8, E, F, W, Z, A_1, faceDEF, faceDEW, faceEFZ
  17. Segment(Midpoint[Vertex[baseTri, 1], Vertex[baseTri, 2]], W) → segTriHeight3
  18. Segment(F, Midpoint[Vertex[baseTri, 1], Vertex[baseTri, 2]]) → triHeight
  19. Midpoint(Vertex[baseTri, 1], Vertex[baseTri, 3]) → LabelTriPos
  20. Textfield(input1) → InputBox1
  21. If(Case ≟ 3, expAns2Tri, expAns2Sq) → expAns2
  22. Segment((0, 1, 0), (1, 1, 0)) → segRad_{1}
  23. Rotate((0, 0, hei_{1}), (pi / 2 * tCil_{1}), Line[(1, 0, 0), (-1, 0, 0)]) → TopPoint
  24. Vector((0, cos((tCil_{1} * pi)), (-sin((tCil_{1} * pi))))) → u
  25. Vector((0, sin((tCil_{1} * pi)), cos((tCil_{1} * pi)))) → v
  26. Corner(5) → App2DSize
  27. TextToUnicode(stringA) → inputCurrent
  28. Translate(f, Vector[(Vector[Corner[4], Corner[1]] / 500 * 16)]) → f'
  29. CornerThreeD(-1, 9) → App3DSize
  30. Vector((1; (pi / 2 * tCil2) + (pi / 6 * (1 - tCil2)); ((-pi)) / 2 + ((0.8 * pi / 2) * (1 - tCil2)))) → view
  31. CornerThreeD(-1, 13) → A
  32. Distance(Corner[text7, 2], Corner[text7, 1]) → widLabelSide
  33. Distance(Corner[text7, 4], Corner[text7, 1]) → heiLabelSide
  34. Translate(F, Vector[(x(Z) + 0.1, 0, 0)]) → L'
  35. Translate(LabelTriPos, Vector[(x(Z) + 0.1, 0, 0)]) → LabelTriPos'
  36. Text(randomSideLength, LabelSqPos + (0, 0.25, (0.25 * tCil2)), true, true) → text6
  37. Translate(segTriHeight4, Vector[(x(L_1) + 0.21, 0, 0)]) → segTriHeight4'
  38. Text(triHeiNum4, Translate[Midpoint[segTriHeight4], Vector[(x(L_1) + 0.25, 0, 0)]], true, true) → LabelHeight4
  39. Segment(If[Case ≟ 3, (0.5, 0.71, 0), (0.5, 0, 0)], Translate[If[Case ≟ 3, (0.5, 0.71, 0), (0.5, 0, 0)], Vector[(0, 0, hei_{1})]]) → PyrHei
  40. Text(hei2, Midpoint[PyrHei], true, true) → text5
  41. Text(randomSideLength, LabelTriPos + ((-widLabelSide), heiLabelSide, 0), true, true) → text4
  42. Vector(Midpoint[PyrHei], Vertex[PyrHei, 1]) → PyrHei2
  43. Vector(Midpoint[PyrHei], Vertex[PyrHei, 2]) → PyrHei1
  44. Translate(Midpoint[triHeight], Vector[((x(Z) * 1.1), 0)]) → G
  45. Translate(F, Vector[((x(Z) * 1.1), 0)]) → F'
  46. Vector(G, F') → triBaseHei1
  47. Vector(G, G - triBaseHei1) → triBaseHei2
  48. Text(triHeightNum, G + (0, heiLabelSide / 2), true, true) → text8
  49. Translate(Midpoint[Midpoint[Vertex[baseTri, 1], Vertex[baseTri, 2]], W], Vector[(0.75, 0)]) → H
  50. Translate(Midpoint[Vertex[baseTri, 1], Vertex[baseTri, 2]], (0.75, 0)) → I
  51. Translate(W, Vector[(0.75, 0)]) → W'
  52. Vector(H, I) → triFaceHei2
  53. Vector(H, W') → triFaseHei1
  54. Text(triHeiNum3, H, true, true) → LabelHeight3
  55. Vector(PM, P1) → w
  56. Vector(PM, P2) → c
  57. Pyramid(baseSqare, (-hsqHint)) → pyrSqHint, J, face5, face7, face8, face6, edge1, edge2, edge4, edge7
  58. Pyramid(baseTri, hTriHint) → pyrTriHint, K, face9, face11, face10, edge12, edge14, edge15
  59. Net(pyrSqHint, If[hsqHint ≤ hei_{1}, tCil2^(2), sqrt(tCil2)]) → g, L, faceLMP, faceMNQ, faceNOR, faceLOS, edgeLM, edgeMN, edgeNO, edgeLO, edgeLP, edgeMP, M, edgeMQ, edgeNQ, edgeNR, edgeOR, edgeOS, edgeLS, N, O, P, Q, R, S, faceLMNO
  60. Net(pyrTriHint, If[hTriHint ≤ hei_{1}, tCil2^(2), sqrt(tCil2)]) → h, B_1, face23, edge9, edge17, edge10, edge13, edge18, edge19, edge20, edge21, edge16, C_1, D_1, I_1, K_1, M_1, face12, face13, face24
  61. Segment(V_1, P2) → i
  62. Segment(Vertex[edge36, 1], P1) → j
  63. Segment((0, 0.5, hei_{1}), Vertex[PyrHei, 2]) → k
  64. Segment(W, Vertex[PyrHei, 2]) → l
  65. Segment(W, W') → m
  66. Segment(E, I) → n
  67. Line((0, 0, 0), Vector[((n * (e * (m * (g * (e * x(S)))))), (n * (e * (m * (g * (e * y(S)))))), (n * (e * (m * (g * (e * z(S)))))))]) → p
  68. Segment(F, F') → q
  69. Segment(E, G - triBaseHei1) → r

AI 绘图提示语参考

画一个带底面的正四棱锥的空间几何图：先在三维坐标系中给出一个正方形底面，并在底面上定义锥尖，使锥高为可调参数。底面四条边各作为棱，用多面体面把锥体完整表示出来。接着在同一画面中展开并展示该锥体的展开网：以底面向外“展开”为一个平面构型，把四个侧面三角形围绕底边依次放置成网格状。用可计算的量标注底面边长与锥高，并在展开网的对应三角形处给出侧面三角形的高与底边信息；同时添加必要的测量辅助线与中点位置，用于定位三角形的高度标注。最后在网与原锥之间保持对应关系的视觉一致性，包含底面面积、侧面积与总面积的展示空间（用文字标注对应数值位置）。
以上提示语为根据原始绘图命令整理而来，仅供参考。