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圆环的周长与面积：半径差平方与公式应用

圆环的周长与面积：半径差平方与公式应用
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2024-12-18 20:44
所属专题 Kreis Umfang und Flächeninhalt
该范例通过给定内外两个圆的半径，展示圆环（同心环形区域）的面积与周长如何计算，并配套选择题验证可用公式。核心数学概念包括：圆环的面积 = 外圆面积−内圆面积（π(r1^2−r2^2)），圆环的周长由两条边界圆的周长之和构成（2πr1+2πr2），以及在题目中要求将结果按指定小数位数取整。作图与交互要点可从构造摘要提炼：使用 Sequence 生成一组随机参数；用 Circle 创建两个圆；Text 与 Textfield 用于题干与符号展示（如 r1、r2、面积/周长公式选项）；用 Join、Segment 连接相关元素并呈现界面/反馈；用 SelectedIndex 与 If 判断多选答案或步骤；通过 Button 与输入组件（btnBack、btnNew、btnNext、inputBox、showsolution、attemptCount、questionCount、validate）

GeoGebra 命令及参数

  1. Sequence(Random[1, 10], i, 1, 10) → randomNumbers, {8, 2, 8, 7, 3, 2, 9, 10, 5, 4}
  2. Element(randomNumbers, Mod[questionCount, Length[randomNumbers]] + 1) → randomNumber
  3. Textfield(input) → inputBox
  4. 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
  5. 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
  6. Sequence(Sequence[Corner[4] + (pixMat * ((k * stepXY), (m * stepXY))), k, 0, maxX / stepXY], m, 0, maxY / stepXY) → PR
  7. Segment(MP01, MP01 + (KreisRadius011, 0)) → segR011
  8. Segment(MP01, MP01 + (0, (-KreisRadius012))) → segR012
  9. Segment(MP091, MP091 + (KreisRadius091, 0)) → segR091
  10. Segment(MP091, MP091 + (0, (-KreisRadius092))) → segR092
  11. Segment(MP092, MP092 + (0, (-KreisRadius092))) → segR094
  12. Segment(MP092, MP092 + (KreisRadius091, 0)) → segR093
  13. SelectedIndex(dropdown031) → index031
  14. SelectedIndex(dropdown032) → index032
  15. SelectedIndex(dropdown071) → index071
  16. If(step ≟ 3, correctdd031 ∧ correctdd032, step ≟ 7, correctdd071, step ≟ 4, PDD041 ≟ PDDEnd042 ∧ PDD042 ≟ PDDEnd041, step ≟ 8, PDD081 ≟ PDDEnd082 ∧ PDD082 ≟ PDDEnd081, true) → clickableBtnNext
  17. Text("\textsf{\bold{Zieh mich!}}", Midpoint[ParkPDD041, ParkPDD042] + (0, -0.5), false, true, 0, -1) → CapDD041
  18. Text("\textsf{\pi}", PDD081 + (0, 0.05), false, true, 0, 0) → txtDD081
  19. Text("\textsf{2\:r_1}", PDD082 + (0, 0.05), false, true, 0, 0) → txtDD082
  20. Text("\textsf{\bold{Zieh mich!}}", Midpoint[ParkPDD081, ParkPDD082] + (0, -0.5), false, true, 0, -1) → CapDD081
  21. Circle(MP01, KreisRadius011) → KreisSchraffierung1
  22. Circle(MP01, KreisRadius012) → KreisSchraffierung2
  23. Text("\textsf{r_2²}", PDD041 + (0, 0.05), false, true, 0, 0) → txtDD041
  24. Text("\textsf{r_1²}", PDD042 + (0, 0.05), false, true, 0, 0) → txtDD042
  25. Circle(MP092, KreisRadius011) → KreisArea093
  26. Circle(MP092, KreisRadius012) → KreisArea094