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圆柱表面积公式

圆柱表面积公式
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2023-08-11 03:10
立体几何几何作图
这个范例通过展开圆柱的网来演示圆柱的表面积计算。涉及的核心数学概念包括圆柱的侧面积和表面积公式。作图要点使用了GeoGebra的Sequence, Shuffle, Element, Zip, Textfield, If, Segment, Circle, Polygon, Corner, Rotate, Vector, Translate, Cylinder, Midpoint, Text等命令和对象。适合6-8年级学生学习如何使用公式计算圆柱的表面积。

GeoGebra 命令及参数

  1. Sequence(5) → ListRadius_{1}
  2. Shuffle(ListRadius_{1}) → randomListRadius_{1}, {5, 1, 4, 2, 3}
  3. Element(randomListRadius_{1}, Mod[countQuestion_{1}, Length[randomListRadius_{1}]] + 1) → randomRadius
  4. Zip(If[Mod[kk, 3] ≟ 0, Sequence[1 / 3, 5, 1 / 3], If[Mod[kk, 2] ≟ 0, Sequence[1 / 2, 5, 1 / 2], Sequence[5]]], kk, randomListRadius_{1}) → ListHeights
  5. Shuffle(ListHeightsC) → randomListHeight_{1}, {1, 4, 5, 2, 3}
  6. Textfield(input1) → InputBox1
  7. Shuffle(Sequence[0.1, 2, 0.01]) → randomListIncrement_{1}, {0.9400000000000007, 1.11, 0.8600000000000007, 0.43000000000000027, 0.16000000000000003, 0.4600000000000003, 0.41000000000000025, 0.2600000000000001, 1.7900000000000007, 1.1300000000000001, 0.8500000000000006, 0.9300000000000007, 0.31000000000000016, 1.1900000000000002, 0.20000000000000007, 1.05, 0.6700000000000005, 1.3800000000000003, 1.4900000000000004, 1.4700000000000004, 1.8500000000000008, 0.6600000000000005, 1.9600000000000009, 0.9500000000000007, 0.15000000000000002, 1.9900000000000009, 1.9100000000000008, 0.3400000000000002, 1.6900000000000006, 1.01, 0.11, 1.04, 1.7400000000000007, 0.5200000000000004, 0.5400000000000004, 1.2000000000000002, 0.6500000000000005, 1.4400000000000004, 0.19000000000000006, 0.42000000000000026, 0.5800000000000004, 1.8000000000000007, 1.1800000000000002, 0.4400000000000003, 1.06, 1.7700000000000007, 1.7100000000000006, 0.4500000000000003, 1.3600000000000003, 1.5600000000000005, 0.3500000000000002, 0.9200000000000007, 0.8300000000000006, 0.7200000000000005, 0.7300000000000005, 0.4900000000000003, 0.28000000000000014, 0.7600000000000006, 1.6800000000000006, 1, 1.4800000000000004, 1.7200000000000006, 0.5900000000000004, 0.39000000000000024, 1.6000000000000005, 0.1, 1.9300000000000008, 0.9900000000000008, 1.1400000000000001, 1.2600000000000002, 0.18000000000000005, 0.5600000000000004, 1.5300000000000005, 1.8900000000000008, 0.6400000000000005, 1.8800000000000008, 1.9400000000000008, 1.5100000000000005, 0.6100000000000004, 1.2100000000000002, 0.29000000000000015, 1.4100000000000004, 1.2900000000000003, 1.7000000000000006, 1.03, 0.2400000000000001, 0.9100000000000007, 1.8700000000000008, 0.2300000000000001, 1.7600000000000007, 1.7800000000000007, 0.8400000000000006, 1.5800000000000005, 1.8300000000000007, 0.7800000000000006, 1.7500000000000007, 1.6500000000000006, 1.08, 0.3300000000000002, 1.4200000000000004, 0.30000000000000016, 0.8700000000000007, 0.22000000000000008, 0.4700000000000003, 0.4800000000000003, 1.9700000000000009, 1.02, 1.5700000000000005, 0.8000000000000006, 1.8400000000000007, 1.2300000000000002, 1.1, 1.5400000000000005, 1.5500000000000005, 1.9800000000000009, 1.2400000000000002, 1.6700000000000006, 1.3400000000000003, 1.8600000000000008, 0.13, 1.8200000000000007, 1.7300000000000006, 1.3100000000000003, 1.2200000000000002, 1.1700000000000002, 1.3300000000000003, 0.7100000000000005, 0.17000000000000004, 1.4300000000000004, 0.7700000000000006, 0.14, 1.9500000000000008, 1.4500000000000004, 1.3200000000000003, 0.6200000000000004, 0.9600000000000007, 0.5300000000000004, 1.6200000000000006, 0.3200000000000002, 1.2800000000000002, 1.12, 0.6300000000000004, 1.1600000000000001, 0.9800000000000008, 1.6600000000000006, 2, 1.3700000000000003, 0.3800000000000002, 0.21000000000000008, 0.7900000000000006, 1.6300000000000006, 1.3900000000000003, 1.5000000000000004, 1.5900000000000005, 1.9200000000000008, 1.2500000000000002, 0.6000000000000004, 1.1500000000000001, 0.8100000000000006, 1.3000000000000003, 0.5700000000000004, 1.6100000000000005, 0.6800000000000005, 0.8800000000000007, 0.5100000000000003, 0.2500000000000001, 0.8900000000000007, 1.2700000000000002, 1.9000000000000008, 1.4600000000000004, 1.09, 1.6400000000000006, 1.07, 0.40000000000000024, 0.3600000000000002, 0.9000000000000007, 0.27000000000000013, 0.3700000000000002, 0.6900000000000005, 0.5500000000000004, 1.4000000000000004, 1.8100000000000007, 0.7000000000000005, 1.3500000000000003, 0.7400000000000005, 0.5000000000000003, 0.8200000000000006, 1.5200000000000005, 0.7500000000000006, 0.9700000000000008, 0.12}
  8. Shuffle(Sequence[15]) → randomList2, {14, 2, 4, 9, 12, 8, 7, 15, 11, 6, 10, 1, 5, 3, 13}
  9. Shuffle(Sequence[20]) → randomList1, {13, 14, 2, 17, 1, 5, 20, 18, 4, 8, 11, 7, 12, 3, 6, 15, 16, 19, 9, 10}
  10. Element(randomList2, Mod[countQuestion, 15] + 1) → randomNum2
  11. Element(randomList1, Mod[countQuestion, 20] + 1) → randomNum1
  12. If(abs(tCil_{1}) < 1, ((rCil_{1} * sin((-θCil_{1}))), ((rCil_{1} * (1 - cos((-θCil_{1})))) * cos(ϕCil_{1})) - (hei_{1} * sin(ϕCil_{1})), (hei_{1} * cos(ϕCil_{1})) + ((rCil_{1} * (1 - cos((-θCil_{1})))) * sin(ϕCil_{1}))), ((-pi), ((-hei_{1}) * sin(ϕCil_{1})), (hei_{1} * cos(ϕCil_{1})))) → CE_{1}
  13. If(abs(tCil_{1}) < 1, ((rCil_{1} * sin(θCil_{1})), ((rCil_{1} * (1 - cos(θCil_{1}))) * cos(ϕCil_{1})) - (hei_{1} * sin(ϕCil_{1})), (hei_{1} * cos(ϕCil_{1})) + ((rCil_{1} * (1 - cos(θCil_{1}))) * sin(ϕCil_{1}))), (pi, ((-hei_{1}) * sin(ϕCil_{1})), (hei_{1} * cos(ϕCil_{1})))) → CF_{1}
  14. If(abs(tCil_{1}) < 1, ((rCil_{1} * sin((-θCil_{1}))), ((rCil_{1} * (1 - cos((-θCil_{1})))) * cos(ϕCil_{1})), ((rCil_{1} * (1 - cos((-θCil_{1})))) * sin(ϕCil_{1}))), ((-pi), 0, 0)) → CG_{1}
  15. If(abs(tCil_{1}) < 1, ((rCil_{1} * sin(θCil_{1})), ((rCil_{1} * (1 - cos(θCil_{1}))) * cos(ϕCil_{1})), ((rCil_{1} * (1 - cos(θCil_{1}))) * sin(ϕCil_{1}))), (pi, 0, 0)) → CH_{1}
  16. Segment(CE_{1}, CG_{1}) → mCil_{1}
  17. Segment(CF_{1}, CH_{1}) → nCil_{1}
  18. If(abs(tCil_{1}) < 1, CurveCartesian[(rCil_{1} * sin((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1}))), ((rCil_{1} * (1 - cos((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1})))) * cos(ϕCil_{1})), ((rCil_{1} * (1 - cos((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1})))) * sin(ϕCil_{1})), CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu, -1, 1], CurveCartesian[(pi * CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu), 0, 0, CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu, -1, 1]) → lCil_{1}
  19. If(abs(tCil_{1}) < 1, CurveCartesian[(rCil_{1} * sin((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1}))), ((rCil_{1} * (1 - cos((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1})))) * cos(ϕCil_{1})) - (hei_{1} * sin(ϕCil_{1})), (hei_{1} * cos(ϕCil_{1})) + ((rCil_{1} * (1 - cos((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1})))) * sin(ϕCil_{1})), CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu, -1, 1], CurveCartesian[(pi * CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu), ((-hei_{1}) * sin(ϕCil_{1})), (hei_{1} * cos(ϕCil_{1})), CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu, -1, 1]) → jCil_{1}
  20. If(abs(tCil_{1}) < 1, Surface[(rCil_{1} * sin((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1}))), ((rCil_{1} * (1 - cos((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1})))) * cos(ϕCil_{1})) - (CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGv * sin(ϕCil_{1})), (CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGv * cos(ϕCil_{1})) + ((rCil_{1} * (1 - cos((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1})))) * sin(ϕCil_{1})), CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu, -1, 1, CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGv, 0, hei_{1}], Surface[(pi * CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu), ((-CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGv) * sin(ϕCil_{1})), (CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGv * cos(ϕCil_{1})), CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu, -1, 1, CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGv, 0, hei_{1}]) → iCil_{1}
  21. Circle(BaseCenter_{1}, 1, Vector[(0, 0, 1)]) → BaseCil2
  22. Segment((0, 1, 0), (1, 1, 0)) → segRad_{1}
  23. Polygon({((-pi), 0, 0), (pi, 0, 0), (pi, (-hei_{1}), 0), ((-pi), (-hei_{1}), 0)}) → unfoldedCilinder_{1}
  24. Corner(5) → L_{1}
  25. Circle(BaseCenter_{1}, 1, Vector[(0, 0, 1)]) → BaseCil
  26. Circle(Translate[BaseCenter_{1}, Vector[(0, 0, hei_{1})]], 1, Vector[(0, 0, 1)]) → TopCilFixed
  27. Rotate((0, 0, hei_{1}), (pi / 2 * tCil_{1}), Line[(1, 0, 0), (-1, 0, 0)]) → TopPoint
  28. Vector((0, cos((tCil_{1} * pi)), (-sin((tCil_{1} * pi))))) → u
  29. Translate(TopPoint, u) → TopPointC
  30. Vector((0, sin((tCil_{1} * pi)), cos((tCil_{1} * pi)))) → v
  31. Circle(TopPointC, 1, v) → TopCil
  32. Corner(5) → A
  33. Cylinder(baseHint, heiHint) → a, topHint, b
  34. Midpoint(CG_{1}, CE_{1}) → labelPos
  35. Vector((1; (pi / 2 * tCil2); ((-pi)) / 2 + ((0.8 * pi / 2) * (1 - tCil2)))) → view
  36. Translate(f, Vector[(Vector[Corner[4], Corner[1]] / 500 * 16)]) → f'
  37. Rotate(TopPoint, 90°, TopPointC, v) → TopPoint'
  38. Segment(TopPoint', TopPointC) → segRadTop
  39. Text(labelRadius_{1}, TopPointC + ((0.2 * tCil2), (0.5 * tCil2), (tCil2 * hei_{1}) + 1), true, true) → text4

AI 绘图提示语参考

生成一个圆柱体的三维视图，并同时给出把圆柱侧面展开后的平面网。先建立圆柱底面与顶面圆，圆柱侧表面对应的生成面以高度为参数连接；在侧面上取一条由圆周展开位置对应的曲线边界，与两条与展开边界相邻的竖直边界用线段封合成展开条带。再绘制展开条带的四边形区域：底边与顶边分别为半周展开得到的两条水平边界，左右边界为与高度方向对应的两条竖边；最后添加文字标注，用于显示半径相关信息并放置在展开后的视图区域附近。
以上提示语为根据原始绘图命令整理而来，仅供参考。