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相似三维图形的体积与表面积关系 by Alicia Chilton

相似三维图形的体积与表面积关系 by Alicia Chilton
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2018-05-08 08:49
立体几何最值优化几何作图
①本范例演示了相似三维图形（如棱柱）的边长、表面积与体积之间的关系；②涉及核心数学概念包括相似比、比例关系、体积与表面积的计算公式；③作图要点——使用 GeoGebra 命令 Polygon（多边形）、Prism（棱柱）、Point（点）、Segment（线段）、Rotate（旋转）、Intersect（交点）、Vector（向量）等构建图形，并通过对象 A, l, w, h, B, C, D, E, poly1, a, b, c, d, e, F, G, H, faceABFE, faceBCGF, faceCDHG, faceADHE, faceEFGH, edgeAE, edgeBF, edgeCG, edgeDH, edgeEF, edgeFG, edgeGH, edgeEH, A', E', k, B', C', D', poly2, a', b', c' 进行具体演示；④适合高中阶段学习立体几何的学生，用于理解相似图形的数学性质，并应用于实际计算问题。

GeoGebra 命令及参数

  1. Polygon(A, B, C, D) → poly1, a, b, c, d
  2. Prism(poly1, E) → e, F, edgeBF, edgeCG, edgeDH, edgeEF, edgeFG, edgeGH, edgeEH, G, H, faceABFE, faceBCGF, faceCDHG, faceADHE, faceEFGH, edgeAE
  3. Point(Segment[A' + (0, 0, h), A' + (0, 0, (2 * h))]) → E'
  4. Polygon(A', B', C', D') → poly2, a', b', c', d'
  5. Prism(poly2, E') → f, F', edgeB'I, edgeC'J, edgeD'K, edgeE'I, edgeIJ, edgeJK, edgeE'K, G', H', faceA'B'IE', faceB'C'JI, faceC'D'KJ, faceA'D'KE', faceE'IJK, edgeA'E'
  6. Segment(I, J) → g
  7. Point(g) → K
  8. Rotate(faceABFE, ((t * 90))°, a) → faceABFE'
  9. Rotate(faceBCGF, ((t * 90))°, b) → faceBCGF'
  10. Rotate(faceCDHG, ((t * 90))°, c) → faceCDHG'
  11. Rotate(faceADHE, ((t * 90))°, d) → faceADHE'
  12. Rotate(E, ((t * 90))°, a) → E'_1
  13. Rotate(F, ((t * 90))°, a) → F'_1
  14. Rotate(Rotate[faceEFGH, ((t * 90))°, a], ((t * 90))°, Segment[E'_1, F'_1]) → poly3
  15. Rotate(faceA'B'IE', ((t * 90))°, a') → faceA'B'IE''
  16. Rotate(faceB'C'JI, ((t * 90))°, b') → faceB'C'JI'
  17. Rotate(faceC'D'KJ, ((t * 90))°, c') → faceC'D'KJ'
  18. Rotate(faceA'D'KE', ((t * 90))°, d') → faceA'D'KE''
  19. Rotate(E', ((t * 90))°, a') → E''
  20. Rotate(F', ((t * 90))°, a') → F''
  21. Rotate(Rotate[faceE'IJK, ((t * 90))°, a'], ((t * 90))°, Segment[E'', F'']) → poly4
  22. Rotate(Segment[Midpoint[B', C'], Midpoint[F', G']], ((t * 90))°, b') → j
  23. Rotate(Segment[Midpoint[B', F'], Midpoint[C', G']], ((t * 90))°, b') → i
  24. Rotate(Segment[Midpoint[A', B'], Midpoint[E', F']], ((t * 90))°, a') → m
  25. Rotate(Segment[Midpoint[A', E'], Midpoint[B', F']], ((t * 90))°, a') → n
  26. Rotate(Segment[Midpoint[C', D'], Midpoint[G', H']], ((t * 90))°, c') → p
  27. Rotate(Segment[Midpoint[C', G'], Midpoint[D', H']], ((t * 90))°, c') → q
  28. Rotate(Segment[Midpoint[A', E'], Midpoint[D', H']], ((t * 90))°, d') → r
  29. Rotate(Segment[Midpoint[A', D'], Midpoint[E', H']], ((t * 90))°, d') → s
  30. Rotate(Rotate[Segment[Midpoint[E', F'], Midpoint[G', H']], ((t * 90))°, a'], ((t * 90))°, Segment[E'', F'']) → a_1
  31. Rotate(Rotate[Segment[Midpoint[E', H'], Midpoint[G', F']], ((t * 90))°, a'], ((t * 90))°, Segment[E'', F'']) → b_1
  32. Segment(Midpoint[A', B'], Midpoint[C', D']) → c_1
  33. Segment(Midpoint[A', D'], Midpoint[C', B']) → d_1
  34. Segment(Midpoint[F', C'], Midpoint[E', D']) → e_1
  35. Segment(Midpoint[F', A'], Midpoint[C', H']) → f_1
  36. Intersect(Line[E, E'], z = 0) → L
  37. Segment(L, E) → g_1
  38. Segment(L, H) → i_1
  39. Segment(L, F) → k_1
  40. Segment(L, G) → m_1
  41. Segment(H, H') → j_1
  42. Segment(G, G') → n_1
  43. Segment(E, E') → h_1
  44. Segment(F, F') → l_1
  45. Segment(L, A) → p_1
  46. Segment(A, A') → q_1
  47. Segment(L, B) → r_1
  48. Segment(B, B') → s_1
  49. Segment(L, C) → t_1
  50. Segment(C, C') → a_2
  51. Segment(L, D) → b_2
  52. Segment(D, D') → c_2
  53. Vector(E' + (0, 0, 0.3), E' + (0, 0, 1.3)) → u
  54. Segment(Midpoint[A', C'], Midpoint[E', G']) → d_2