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Volumes and Surface Areas of Similar 3D Figures by powere

Volumes and Surface Areas of Similar 3D Figures by powere
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2018-08-14 02:13
Solid GeometryGeometric Construction
This example demonstrates the relationships between side lengths, surface areas, and volumes of similar 3D figures, such as prisms. Key math concepts include similarity ratios, proportional relationships, and volume/surface area formulas. GeoGebra commands and objects used in construction are Polygon, Prism, Point, Segment, Rotate, Intersect, and Vector. Suitable for high school students learning solid geometry to visually understand the quantitative relationships in similar shapes.

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

AI Drawing Prompt Reference

Draw two similar 3D prisms, one smaller and one larger, to illustrate the relationships between side lengths, surface areas, and volumes. Start with a quadrilateral-based prism, then construct a larger similar prism above it, sharing the same central axis. Use semi-transparent faces to show the internal structure. Add a rotation animation around the base edges to view from different angles. Draw segments connecting corresponding vertices of the two prisms, and auxiliary lines from the base center to the top vertices to highlight the similarity ratio. Label key dimensions and ratios, such as side length ratio, surface area ratio, and volume ratio, using different colors for clarity.
The prompt above was compiled from the original drawing commands and is for reference only.