积极 AI Logo 积极 AI

金字塔体积公式的演示

金字塔体积公式的演示
几何画板及绘图内容加载中，请稍等……
2017-07-17 12:04
立体几何几何作图
①本范例通过动态演示证明金字塔体积公式：体积等于底面积乘以高再除以三；②涉及核心数学概念包括体积计算、几何体构造、比例关系；③构造过程中运用了If、Translate、Intersect、Pyramid、Dilate、Polygon等GeoGebra命令与对象，如通过Translate平移、Intersect交点、Pyramid生成金字塔等；④适合高中立体几何教学，用于直观理解体积公式的推导过程。

GeoGebra 命令及参数

  1. If(s ≤ 0, (-s), 0) → s2
  2. If(s ≥ 0, s, 0) → s1
  3. Translate(C, Vector[B, D]) → C'
  4. Intersect(OrthogonalPlane[B, Segment[A, C]], Line[A, C]) → G
  5. Translate(A, Vector[B, D]) → A'
  6. Intersect(OrthogonalPlane[B, Segment[A, C]], Line[A', C']) → H
  7. Intersect(OrthogonalPlane[C, Segment[A, B]], Line[A', D]) → F
  8. Intersect(OrthogonalPlane[C, Segment[A, B]], Line[A, B]) → E
  9. Translate(Rotate[Translate[Rotate[C, (α * s1), Segment[A, B]], Vector[(s1 * Vector[(0, 0, 0), (0, z(D), 0)])]], ((-β) * s2), Segment[A, C]], Vector[(s2 / 2 * Vector[(0, 0, 0), (z(D), 0, 0)])]) → L
  10. Translate(Rotate[Translate[Rotate[Polygon[A, D, A'], (α * s1), Segment[A, B]], Vector[(s1 * Vector[(0, 0, 0), (0, z(D), 0)])]], ((-β) * s2), Segment[A, C]], Vector[(s2 / 2 * Vector[(0, 0, 0), (z(D), 0, 0)])]) → poly2
  11. Pyramid(Polygon[A, B, C], D) → a, faceABD, faceBCD, faceACD_1, edgeAD, edgeBD, edgeCD_1
  12. Translate(Rotate[Translate[Rotate[D, (α * s1), Segment[A, B]], Vector[(s1 * Vector[(0, 0, 0), (0, z(D), 0)])]], ((-β) * s2), Segment[A, C]], Vector[(s2 / 2 * Vector[(0, 0, 0), (z(D), 0, 0)])]) → K
  13. Translate(Rotate[Translate[Rotate[Polygon[A, C, A'], (α * s1), Segment[A, B]], Vector[(s1 * Vector[(0, 0, 0), (0, z(D), 0)])]], ((-β) * s2), Segment[A, C]], Vector[(s2 / 2 * Vector[(0, 0, 0), (z(D), 0, 0)])]) → poly3
  14. Pyramid(Polygon[A', D, C'], C) → b, faceA'CD_1, faceCC'D, faceA'CC', edgeA'C_1, edgeCD_2, edgeCC'
  15. Translate(Translate[Rotate[a, (α * s1), Segment[A, B]], Vector[(s1 * Vector[(0, 0, 0), (0, z(D), 0)])]], Vector[(o2 * Vector[A, B])]) → a'
  16. Pyramid(Polygon[A, D, A'], C) → c, faceACD, faceA'CD, faceAA'C, edgeAC, edgeCD, edgeA'C
  17. Translate(Rotate[c, (α * s1), Segment[A, B]], Vector[(s1 * Vector[(0, 0, 0), (0, z(D), 0)])]) → c'
  18. Translate(Rotate[Translate[b, Vector[(o1 * Vector[A, C])]], (s1 * α), Segment[A, B]], Vector[(s1 * Vector[(0, 0, 0), (0, z(D), 0)])]) → b'
  19. Translate(Rotate[a', ((-β) * s2), Segment[A, C]], Vector[(s2 / 2 * Vector[(0, 0, 0), (z(D), 0, 0)])]) → a''
  20. Translate(Rotate[b', ((-β) * s2), Segment[A, C]], Vector[(s2 / 2 * Vector[(0, 0, 0), (z(D), 0, 0)])]) → b''
  21. Translate(Rotate[c', ((-β) * s2), Segment[A, C]], Vector[(s2 / 2 * Vector[(0, 0, 0), (z(D), 0, 0)])]) → c''
  22. Translate(Rotate[Translate[Translate[Rotate[Polygon[A, B, D], (α * s1), Segment[A, B]], Vector[(s1 * Vector[(0, 0, 0), (0, z(D), 0)])]], Vector[(o2 * Vector[A, B])]], ((-β) * s2), Segment[A, C]], Vector[(s2 / 2 * Vector[(0, 0, 0), (z(D), 0, 0)])]) → poly1
  23. Translate(Rotate[Translate[Translate[Rotate[C, (α * s1), Segment[A, B]], Vector[(s1 * Vector[(0, 0, 0), (0, z(D), 0)])]], Vector[(o2 * Vector[A, B])]], ((-β) * s2), Segment[A, C]], Vector[(s2 / 2 * Vector[(0, 0, 0), (z(D), 0, 0)])]) → I
  24. Dilate(poly2, 1 - h, L) → poly2'
  25. Dilate(poly1, 1 - h, I) → poly1'
  26. Translate(Rotate[Translate[Rotate[Translate[Polygon[A', C, C'], Vector[(o1 * Vector[A, C])]], (s1 * α), Segment[A, B]], Vector[(s1 * Vector[(0, 0, 0), (0, z(D), 0)])]], ((-β) * s2), Segment[A, C]], Vector[(s2 / 2 * Vector[(0, 0, 0), (z(D), 0, 0)])]) → poly4
  27. Translate(Rotate[Translate[Rotate[Translate[D, Vector[(o1 * Vector[A, C])]], (s1 * α), Segment[A, B]], Vector[(s1 * Vector[(0, 0, 0), (0, z(D), 0)])]], ((-β) * s2), Segment[A, C]], Vector[(s2 / 2 * Vector[(0, 0, 0), (z(D), 0, 0)])]) → J
  28. Dilate(poly3, 1 - h, K) → poly3'
  29. Dilate(poly4, 1 - h, J) → poly4'
  30. Polygon((-6, -6), (6, -6), (6, 6), (-6, 6)) → poly5, d, e, f, g