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长方体体积公式推导：已知底面积与高度求体积

长方体体积公式推导：已知底面积与高度求体积
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2023-03-23 22:41
立体几何几何作图滑块动画
所属专题 Quader und Würfel - Volumen
本范例演示如何利用底面积G和高度c计算长方体体积。核心数学概念是体积公式V=G×c的应用，以及从单位立方体分割推导体积公式的过程。构造要点：使用Sequence命令生成三维立方体网格展示cubesHorizontal和cubesVertical，配合SpeechBubbleLeft/Right呈现学生对话（Selina、Jule、Stef的讨论），通过Textfield和input输入框实现交互式计算练习。第二个问题拓展到立方体已知底面积（9cm²）时如何推导边长并计算体积。适合5-6年级学生理解体积概念、掌握公式应用、培养空间观念。

GeoGebra 命令及参数

  1. SpeechBubbleLeft(Pbubble1, 9, 1) → bubble1
  2. SpeechBubbleRight(Pbubble2, 6.7, 1.5) → bubble2
  3. Segment(A, B) → ab
  4. Segment(A, E) → ae
  5. Segment(B, F) → bf
  6. Segment(E, F) → ef
  7. Segment(A, D) → ad
  8. Segment(E, H) → eh
  9. Segment(D, H) → dh
  10. Segment(C, D) → cd
  11. Segment(B, C) → bc
  12. Segment(F, G) → fg
  13. Segment(C, G) → cg
  14. Segment(H, G) → hg
  15. SpeechBubbleLeft(Pbubble1 - (0, 0.1), 9.8, 1.1) → bubble3
  16. Textfield(input1) → InputBox1
  17. Sequence(Sequence[Polygon[A + (i, j), A + (i, j) + (cubeLengthWuerfel / 10, 0), A + (i, j) + (cubeLengthWuerfel / 10, cubeLengthWuerfel / 10), A + (i, j) + (0, cubeLengthWuerfel / 10)], i, 0, cubesHorizontal - 1], j, 0, cubesVertical - 1) → m1, {{0.9999999999999991, 0.9999999999999982, 1, 1}}
  18. Sequence(Sequence[Polygon[A + (i, cubesVertical) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)), A + (i, cubesVertical) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)) + (cubeLengthWuerfel / 10, 0), A + (i, cubesVertical) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)) + (cubeLengthWuerfel / 10 + cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10), A + (i, cubesVertical) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)) + (cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10)], i, 0, cubesHorizontal - 1], j, 0, cubesDepth - 1) → m2, {{0.3535533905932735, 0.35355339059327395, 0.35355339059327395, 0.35355339059327306}, {0.3535533905932744, 0.3535533905932722, 0.3535533905932766, 0.35355339059327306}, {0.35355339059327395, 0.35355339059327306, 0.35355339059327573, 0.3535533905932713}}
  19. Sequence(Sequence[Polygon[A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)), A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)) + (cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10), A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)) + (cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10 + cubeLengthWuerfel / 10), A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)) + (0, cubeLengthWuerfel / 10)], i, 0, cubesDepth - 1], j, 0, cubesVertical - 1) → m3, {{0.3535533905932766, 0.35355339059327306, 0.35355339059327484}}
  20. SpeechBubbleRight(Pbubble3, 11, 1.6) → bubble4
  21. Textfield(input2) → InputBox2
  22. Textfield(input3) → InputBox3
  23. SpeechBubbleLeft(Pbubble1 - (0, 2), 9.8, 3.1) → bubble5
  24. Textfield(input4) → InputBox4
  25. Textfield(input5) → InputBox5
  26. SpeechBubbleRight(Pbubble3, 6.3, 1.2) → bubble6
  27. SpeechBubbleRight(Pbubble3 - (0, 0.5), 11, 2) → bubble7
  28. SpeechBubbleRight(Pbubble3 + (0, 0.4), 11.5, 1.2) → bubble8
  29. Textfield(input6) → InputBox6
  30. Textfield(input7) → InputBox7
  31. Textfield(input8) → InputBox8
  32. Sequence(Sequence[Polygon[A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)), A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)) + (cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10), A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)) + (cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10 + cubeLengthWuerfel / 10), A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)) + (0, cubeLengthWuerfel / 10)], i, 0, cubesDepth - 1], j, 0, 1 - 1) → m9, {{0.3535533905932766, 0.35355339059327306, 0.35355339059327484}}
  33. SpeechBubbleRight(Pbubble3 + (0.2, -0.7), 11.5, 1.5) → bubble9
  34. SpeechBubbleLeft(Pbubble1, 12, 1) → bubble10
  35. Sequence(Sequence[Segment[A + (i, j), A + (i, j) + (cubeLengthWuerfel / 10, 0)], i, 0, cubesHorizontal - 1], j, 1, cubesVertical) → m11, {{0.9999999999999998, 1, 1, 1}}
  36. Sequence(Sequence[Segment[A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)), A + (cubesHorizontal, j) + ((i * cubeDepthCathetersWuerfel / 10), (i * cubeDepthCathetersWuerfel / 10)) + (cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10)], i, 0, cubesDepth - 1], j, 1, cubesVertical) → m12, {{0.5000000000000003, 0.5000000000000003, 0.5000000000000003}}
  37. Polygon(A, E, F, B) → q1, a, e, f, b
  38. Sequence(Sequence[Polygon[A + (i, j), A + (i, j) + (cubeLengthWuerfel / 10, 0), A + (i, j) + (cubeLengthWuerfel / 10, cubeLengthWuerfel / 10), A + (i, j) + (0, cubeLengthWuerfel / 10)], i, 0, 4 - 1], j, 0, 1 - 1) → m7, {{0.9999999999999991, 0.9999999999999982, 1, 1}}
  39. Sequence(Sequence[Polygon[A + (i, 1) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)), A + (i, 1) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)) + (cubeLengthWuerfel / 10, 0), A + (i, 1) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)) + (cubeLengthWuerfel / 10 + cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10), A + (i, 1) + ((j * cubeDepthCathetersWuerfel / 10), (j * cubeDepthCathetersWuerfel / 10)) + (cubeDepthCathetersWuerfel / 10, cubeDepthCathetersWuerfel / 10)], i, 0, cubesHorizontal - 1], j, 0, 4) → m8, {{0.3535533905932735, 0.35355339059327395, 0.35355339059327395, 0.35355339059327306}, {0.3535533905932744, 0.3535533905932722, 0.3535533905932766, 0.35355339059327306}, {0.35355339059327395, 0.35355339059327306, 0.35355339059327573, 0.3535533905932713}, {0.35355339059327306, 0.35355339059327306, 0.35355339059327306, 0.35355339059327306}, {0.35355339059327395, 0.35355339059327484, 0.35355339059327395, 0.3535533905932722}}

AI 绘图提示语参考

画一个立方体，展示其体积公式推导。立方体由多个小立方体组成，通过网格线和小立方体堆叠，直观表达体积等于底面积乘以高度。在立方体周围添加对话气泡，分别标注底面积和高度数值，并展示计算过程和结果。在立方体下方添加输入框，用于交互式输入底面积和高度，计算并显示体积。
以上提示语为根据原始绘图命令整理而来，仅供参考。