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Volumes and Surface Areas of Similar 3D Figures by Alicia Chilton

Volumes and Surface Areas of Similar 3D Figures by Alicia Chilton
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2018-05-08 08:49
Solid GeometryOptimizationGeometric Construction
This example demonstrates the relationships between side lengths, surface areas, and volumes of similar three-dimensional figures, such as prisms. Key math concepts include similarity ratios, proportional relationships, and the formulas for calculating volume and surface area. It is constructed using GeoGebra commands like Polygon, Prism, Point, Segment, Rotate, Intersect, and Vector, with objects like 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'. Suitable for high school students learning solid geometry to understand the mathematical properties of similar figures and apply them to practical calculations.

GeoGebra Commands & Parameters

  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