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金字塔表面积公式

金字塔表面积公式
几何画板及绘图内容加载中，请稍等……
2023-12-29 20:34
多边形立体几何几何作图滑块动画函数图像
这个范例通过展开金字塔的网来演示如何计算金字塔的表面积。涉及的核心数学概念包括表面积公式、多边形面积计算和几何展开。作图要点使用了GeoGebra命令如Polygon、Net、Segment等来构建金字塔的展开图，并利用Centroid、Translate等命令进行定位。适合6-8年级学生学习立体几何中的表面积计算。

GeoGebra 命令及参数

  1. RandomUniform(0, 0.99, 200) → allRandomSeeds, {0.6122159166703468, 0.3564662340165225, 0.7058117394068154, 0.6898203120627828, 0.36617096876571625, 0.22692246643669758, 0.06064277277426987, 0.6500166690780637, 0.5658387665973428, 0.293676503652412, 0.8760384328954288, 0.5722726516817781, 0.5333762147521898, 0.4519651137910648, 0.39483074164010495, 0.16954460492333584, 0.2944674120093246, 0.11786328575632649, 0.11434966406767379, 0.9228652727278263, 0.6417895056997956, 0.2405363505049711, 0.34864135284917186, 0.25909369485609407, 0.8188525369174884, 0.6010559411857451, 0.5438919840527319, 0.7405589431598505, 0.9616213948124592, 0.6404788444116846, 0.07901055034690437, 0.45411632774996336, 0.9134041925040105, 0.41288519497951975, 0.43311998528944606, 0.8306001486316313, 0.24750487091948922, 0.7505083939714439, 0.43743799387993404, 0.34256275950057524, 0.7520895889862098, 0.5379620363998807, 0.3606984499301604, 0.25853026320504113, 0.17260427100771933, 0.9508798160010301, 0.6848729012960638, 0.21578555908588498, 0.10890772645399774, 0.14103196107108137, 0.5714949674439594, 0.8509053001185654, 0.9470475817851272, 0.10516814939735866, 0.888374268573395, 0.6879322952860513, 0.769308364568912, 0.8684590429267793, 0.23190568081521434, 0.6581563692756885, 0.8707988427682417, 0.8754695095136393, 0.03153742809142836, 0.1430060246137156, 0.2864224511451724, 0.19209488794797686, 0.6302560720617569, 0.11252016043753511, 0.9385882374505089, 0.9108265216206712, 0.9181795584395868, 0.0062732922150543526, 0.10347088253469387, 0.1890866992410004, 0.7555187899979335, 0.31146943863891896, 0.022016680932895754, 0.38618285251369144, 0.5262757005918627, 0.36419946445505585, 0.8212193870593052, 0.978679512813415, 0.6650912054547105, 0.8245729962054651, 0.061452409437084105, 0.18667173181697, 0.15663806339275896, 0.38459080252833, 0.20684608724089049, 0.754566973742778, 0.579874438719467, 0.7092149077455364, 0.35783654973225193, 0.9195427828925491, 0.6683304903544455, 0.416123309715073, 0.8780894066794313, 0.8388456899678998, 0.7223962419885823, 0.7281027611529458, 0.892015018998223, 0.31353243108513623, 0.8103102880003873, 0.91393768048667, 0.19404316499658322, 0.571127540773592, 0.2312199315360119, 0.05899443807942073, 0.19671374053205762, 0.6729819067904801, 0.0762897819523805, 0.5154804209609335, 0.47146626853214557, 0.753864822301003, 0.6623814794761934, 0.8345732423060717, 0.2837745774533905, 0.6815080594467164, 0.827309469072884, 0.788270432031085, 0.6729742163908746, 0.9624427768042366, 0.03609699670485364, 0.43790110437040247, 0.6867690362002596, 0.547531435124427, 0.714015375270054, 0.668771102540256, 0.2609792468170897, 0.036094336352832726, 0.2644827685285775, 0.236379299450448, 0.5566069870731598, 0.2314662806092862, 0.33813247990129247, 0.4890473439710337, 0.618746896700131, 0.40319167933333044, 0.5535301114470847, 0.2846467977945426, 0.27525747574729614, 0.44187727332855575, 0.8501263369378556, 0.09538731160276294, 0.5774355115212031, 0.31196929030407017, 0.11007713146225465, 0.26434539703138704, 0.2401990721647817, 0.4954076768956765, 0.08754558732278522, 0.5605687082988416, 0.6554865006728863, 0.8552332174454262, 0.07201961549746938, 0.4461818220335911, 0.6597166020684414, 0.2527833454444021, 0.06603685096620171, 0.4289489806883071, 0.1742321188723171, 0.47511124028785295, 0.7675033059510349, 0.6672161998319754, 0.5066618034779392, 0.7206976822108371, 0.27252478869919494, 0.4510593702886258, 0.37108798588941555, 0.3713986493360098, 0.556339576061083, 0.5138065863046527, 0.9557946869852126, 0.13688192390230175, 0.9575490660408182, 0.32188256452239505, 0.14919788504685816, 0.9245542248780775, 0.7743869110032852, 0.18813946108732899, 0.06732322330676097, 0.4231655538923185, 0.2812508246975527, 0.8844932537042124, 0.05989745525111991, 0.18190121248750127, 0.7591401902705539, 0.793775594514079, 0.3767583066194847, 0.5288314591911114, 0.10500044528227193, 0.7582417888909846, 0.6618646341825009, 0.7530879636430979, 0.8422701541278397, 0.05314109239391603, 0.808701049159869, 0.9540706858634374, 0.8181590227702022, 0.35816025404800805}
  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) * 4)) + 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. Segment((0, 1, 0), (1, 1, 0)) → segRad_{1}
  22. Rotate((0, 0, hei_{1}), (pi / 2 * tCil_{1}), Line[(1, 0, 0), (-1, 0, 0)]) → TopPoint
  23. Vector((0, cos((tCil_{1} * pi)), (-sin((tCil_{1} * pi))))) → u
  24. Vector((0, sin((tCil_{1} * pi)), cos((tCil_{1} * pi)))) → v
  25. Corner(5) → App2DSize
  26. Translate(f, Vector[(Vector[Corner[4], Corner[1]] / 500 * 16)]) → f'
  27. CornerThreeD(-1, 9) → App3DSize
  28. Vector((1; (pi / 2 * tCil2) + (pi / 6 * (1 - tCil2)); ((-pi)) / 2 + ((0.8 * pi / 2) * (1 - tCil2)))) → view
  29. CornerThreeD(-1, 13) → A
  30. Distance(Corner[text7, 2], Corner[text7, 1]) → widLabelSide
  31. Distance(Corner[text7, 4], Corner[text7, 1]) → heiLabelSide
  32. Translate(F, Vector[(x(Z) + 0.1, 0, 0)]) → L'
  33. Translate(LabelTriPos, Vector[(x(Z) + 0.1, 0, 0)]) → LabelTriPos'
  34. Text(randomSideLength, LabelSqPos + (0, 0.25, (0.25 * tCil2)), true, true) → text6
  35. Translate(segTriHeight4, Vector[(x(L_1) + 0.21, 0, 0)]) → segTriHeight4'
  36. Text(triHeiNum4, Translate[Midpoint[segTriHeight4], Vector[(x(L_1) + 0.25, 0, 0)]], true, true) → LabelHeight4
  37. Segment((0.5, 0, 0), (0.5, 0, hei_{1})) → PyrHei
  38. Text(hei2, Midpoint[PyrHei], true, true) → text5
  39. Text(randomSideLength, LabelTriPos + ((-widLabelSide), heiLabelSide, 0), true, true) → text4
  40. Vector(Midpoint[PyrHei], Vertex[PyrHei, 1]) → PyrHei2
  41. Vector(Midpoint[PyrHei], Vertex[PyrHei, 2]) → PyrHei1
  42. Translate(Midpoint[triHeight], Vector[(x(Z), 0)]) → G
  43. Translate(F, Vector[(x(Z), 0)]) → F'
  44. Vector(G, F') → triBaseHei1
  45. Vector(G, G - triBaseHei1) → triBaseHei2
  46. Text(triHeightNum, G + (0, heiLabelSide / 2), true, true) → text8
  47. Translate(Midpoint[Midpoint[Vertex[baseTri, 1], Vertex[baseTri, 2]], W], Vector[(0.75, 0)]) → H
  48. Translate(Midpoint[Vertex[baseTri, 1], Vertex[baseTri, 2]], (0.75, 0)) → I
  49. Translate(W, Vector[(0.75, 0)]) → W'
  50. Vector(H, I) → triFaceHei2
  51. Vector(H, W') → triFaseHei1
  52. Text(triHeiNum3, H, true, true) → LabelHeight3
  53. Vector(PM, P1) → w
  54. Vector(PM, P2) → c
  55. Segment(A_1, Vertex[PyrHei, 2]) → h
  56. Segment(Vertex[baseTri, 3], F') → i
  57. Segment(Vertex[baseTri, 2], G - triBaseHei1) → j
  58. Segment(W, W') → k
  59. Segment((0.5, 0, 0), P1) → l
  60. Segment(V_1, P2) → m
  61. Segment((0, 0.5, hei_{1}), Vertex[PyrHei, 2]) → n
  62. Segment(Centroid[baseTri], Vertex[PyrHei, 1]) → g

AI 绘图提示语参考

绘制一个几何教学图形，展示金字塔的表面积计算方法。画面包含两个金字塔及其展开图：左侧是一个正方形底面的金字塔，右侧是一个三角形底面的金字塔。两个金字塔都以其展开的平面网状形式显示在旁边，展示底面和侧面的面积。正方形金字塔的底边标注随机生成的长度数值，三角形金字塔的三个边也有相应的长度标注。两个金字塔的侧面高度通过线段标注显示。整体布局清晰，展示如何通过展开图来计算金字塔的表面积，适合几何教学演示。
以上提示语为根据原始绘图命令整理而来，仅供参考。