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特殊圆形部分的计算练习

特殊圆形部分的计算练习
几何画板及绘图内容加载中，请稍等……
2024-12-19 16:42
所属专题 Kreis Umfang und Flächeninhalt
本范例通过3道递进难度的题目，练习特殊圆心角对应的扇形面积、弧长及圆周长的计算。核心数学概念包括：圆心角公式、弧长公式L=nπr/180、扇形面积公式S=nπr²/360、圆的周长公式C=2πr和面积公式A=πr²。作图要点：使用GeoGebra的If条件判断实现答案验证，Random命令随机生成题目参数，Sequence和Element从题库中选取题目，Textfield和Text实现输入框与反馈文本显示，Translate处理文本翻译，Split和IndexOf解析输入内容。本练习适合7-8年级学生，可用于课堂巩固或课后自主练习，培养公式推导与应用能力。

GeoGebra 命令及参数

  1. Join(Sequence[Shuffle[{1, 2}], k, 1, 50]) → CaseQList
  2. Random(1, 3, 100) → CaseLevel1List, {2, 2, 1, 1, 2, 1, 1, 3, 2, 2, 1, 2, 2, 3, 3, 1, 2, 1, 2, 3, 1, 3, 1, 2, 1, 1, 1, 3, 2, 2, 3, 3, 2, 3, 3, 1, 2, 2, 2, 1, 1, 2, 3, 2, 2, 3, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 3, 1, 2, 1, 3, 1, 2, 1, 2, 3, 1, 3, 3, 3, 3, 2, 1, 3, 3, 2, 3, 3, 3, 1, 1, 3, 2, 3, 2, 3, 1, 3, 2, 1, 3, 3, 2, 2, 2, 3, 1, 3, 3, 3}
  3. Random(1, 5, 100) → CaseLevel2List, {2, 3, 5, 2, 1, 1, 5, 5, 1, 5, 4, 4, 5, 2, 1, 3, 2, 4, 5, 4, 1, 4, 5, 4, 4, 1, 2, 5, 3, 4, 4, 5, 4, 1, 5, 4, 4, 1, 5, 5, 4, 3, 1, 4, 3, 1, 3, 1, 2, 2, 4, 3, 3, 5, 4, 5, 2, 4, 3, 4, 1, 4, 3, 3, 2, 5, 5, 1, 4, 1, 5, 3, 5, 2, 5, 4, 5, 1, 1, 5, 5, 2, 5, 5, 1, 5, 3, 1, 4, 3, 5, 1, 2, 3, 1, 4, 4, 2, 3, 4}
  4. Random(1, 8, 100) → CaseLevel3List, {5, 7, 2, 2, 3, 8, 5, 2, 6, 8, 5, 6, 8, 8, 2, 5, 8, 3, 3, 1, 6, 7, 4, 7, 8, 6, 2, 3, 7, 3, 4, 8, 4, 7, 6, 8, 2, 6, 3, 1, 1, 2, 5, 1, 7, 2, 8, 7, 1, 3, 5, 3, 8, 2, 7, 4, 2, 2, 1, 7, 7, 8, 1, 2, 1, 3, 7, 1, 3, 6, 1, 2, 3, 4, 1, 8, 5, 8, 4, 3, 3, 1, 3, 3, 5, 3, 4, 3, 1, 8, 3, 5, 2, 1, 8, 6, 7, 5, 7, 3}
  5. If(level ≟ 1, If[CaseLevel1 ≟ 1, 0.5, CaseLevel1 ≟ 2, 0.25, 1 / 8], level ≟ 2, If[CaseLevel2 ≟ 1, 1 / 3, CaseLevel2 ≟ 2, 1 / 6, CaseLevel2 ≟ 3, 1 / 12, CaseLevel2 ≟ 4, 1 / 5, 1 / 10], If[CaseLevel3 ≟ 1, 3 / 4, CaseLevel3 ≟ 2, 3 / 8, CaseLevel3 ≟ 3, 5 / 8, CaseLevel3 ≟ 4, 7 / 8, CaseLevel3 ≟ 5, 2 / 3, CaseLevel3 ≟ 6, 5 / 6, CaseLevel3 ≟ 7, 5 / 9, 7 / 9]) → fac
  6. If(CaseQ ≟ 1, ((Radius^(2) * pi) * fac), (((2 * Radius) * pi) * fac) + (2 * Radius)) → solution
  7. Sequence(Random[1, 10], i, 1, 10) → randomNumbers, {3, 8, 5, 2, 4, 2, 9, 8, 3, 8}
  8. Element(randomNumbers, Mod[questionCount, Length[randomNumbers]] + 1) → randomNumber
  9. Textfield(input) → inputBox
  10. Join({Sequence[Segment[Corner[4] + (pixMat * ((k * stepXY), 0)), Corner[4] + (pixMat * ((k * stepXY), maxY))], k, 0, maxX / stepXY], Sequence[Segment[Corner[4] + (pixMat * (0, (k * stepXY))), Corner[4] + (pixMat * (maxX, (k * stepXY)))], k, 0, maxY / stepXY]}) → Raster
  11. Join({Sequence[Text[UnicodeToLetter[65 + k], Midpoint[Corner[4] + (pixMat * ((k * stepXY), maxY)), Corner[4] + (pixMat * (((k + 1) * stepXY), maxY))], false, true, 0, -1], k, 0, maxX / stepXY - 1], Sequence[Text[("" * k), Midpoint[Corner[4] + (pixMat * (maxX, ((k - 1) * stepXY))), Corner[4] + (pixMat * (maxX, (k * stepXY)))], false, true, 1, 0], k, 1, maxY / stepXY]}) → CapRaster
  12. Sequence(Sequence[Corner[4] + (pixMat * ((k * stepXY), (m * stepXY))), k, 0, maxX / stepXY], m, 0, maxY / stepXY) → PR
  13. If(CaseQ ≟ 1, "Flächeninhalt A", "Umfang u") → txtQ
  14. If(CaseLevel1 ≟ 1, "Halbkreis", CaseLevel1 ≟ 2, "Viertelkreis", "Achtelkreis") → txtLevel1
  15. If(CaseLevel2 ≟ 1, "Drittelkreis", CaseLevel2 ≟ 2, "Sechstelkreis", CaseLevel2 ≟ 3, "Zwölftelkreis", CaseLevel2 ≟ 4, "Fünftelkreis", "Zehntelkreis") → txtLevel2
  16. If(CaseLevel3 ≟ 1, "\dfrac{3}{4}\:-Kreis", CaseLevel3 ≟ 6, "\dfrac{5}{6}\:-Kreis", CaseLevel3 ≟ 7, "\dfrac{5}{9}\:-Kreis", "\dfrac{7}{9}\:-Kreis", CaseLevel3 ≟ 2, "\dfrac{3}{8}\:-Kreis", CaseLevel3 ≟ 3, "\dfrac{5}{8}\:-Kreis", CaseLevel3 ≟ 4, "\dfrac{7}{8}\:-Kreis", CaseLevel3 ≟ 5, "\dfrac{2}{3}\:-Kreis") → txtLevel3
  17. Split(Text[(round(solution, 2) * "")], {""}) → ListSolValue
  18. If(Defined[IndexOf[".", ListSolValue]], Length[ListSolValue] - IndexOf[".", ListSolValue], 0) → numberCommaSolution
  19. Numerator(fac) → numFac
  20. Denominator(fac) → denFac
  21. Translate(imgChilliRot1, Vector[((0.8 * dimChilli), 0)]) → imgChilliRot2
  22. Translate(imgChilliRot1, Vector[(2 * Vector[((0.8 * dimChilli), 0)])]) → imgChilliRot3
  23. Translate(imgChilliGrau1, Vector[(1 * Vector[((0.8 * dimChilli), 0)])]) → imgChilliGrau2
  24. Text((("\textsf{" * If[CaseQ ≟ 1, "cm²\vphantom{cm²}", "cm\vphantom{cm²}"]) * "}"), PosIB + (2.35, -0.225), false, true, 1, 0) → txtUnit
  25. If(CaseQ ≟ 1, (Radius^(2) * pi), ((2 * Radius) * pi)) → wholeCircle
  26. If(CaseQ ≟ 2, ((Radius^(2) * pi) * fac), (((2 * Radius) * pi) * fac) + (2 * Radius)) → wrongSolution
  27. Split(Text[(input * "")], {""}) → ListInputValue
  28. If(Defined[IndexOf[".", ListInputValue]], Length[ListInputValue] - IndexOf[".", ListInputValue], 0) → numberCommaInput
  29. Split(Text[(round(solution, numberCommaInput) * "")], {""}) → ListSolRoundInput
  30. Split(Text[(round(solution, 10) * "")], {""}) → ListSolWhole
  31. IndexOf(".", ListInputValue) → indexCommaInput
  32. IndexOf(".", ListSolWhole) → indexCommaSolution