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Visualizing Similar Cuboids: Surface Area and Volume Scaling

Visualizing Similar Cuboids: Surface Area and Volume Scaling
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2025-03-10 14:39
PolygonSolid GeometryTransformationGeometric Construction
In collections Similar Solids
This example demonstrates how two similar 3D cuboids relate: when the cuboid’s linear dimensions are scaled by a factor, its surface area changes with the square of the factor and its volume with the cube of the factor. Key ideas include similarity in solid geometry, the scale factor from a dilation (homothety), and the resulting dimensional scaling laws for area and volume. Construction in GeoGebra uses Polygon to build the base/outline, Prism to create the 3D solids and their faces, and Dilate to generate the scaled cuboid. Segment and Point define measurable edges and reference points, while Vector and Rotate help with spatial placement and orientation. Face objects (e.g., faceABFE, faceBCGF, faceCDHG, faceADHE, faceEFGH) and edge objects (e.g., edgeAE, edgeBF, edgeCG, edgeDH, edgeEF, edgeFG, edgeGH, edgeEH) are used to organize the 3D structure and compare corresponding parts before and after dilation (A, B, C, D, E vs. A', B', C', D', E'). Suitable for upper secondary and early college lessons on solid geometry, similarity, and visualization of scaling relationships.

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. Dilate(A, k, L) → A'
  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. Segment(L, E) → g_1
  37. Segment(L, H) → i_1
  38. Segment(L, F) → k_1
  39. Segment(L, G) → m_1
  40. Segment(H, H') → j_1
  41. Segment(G, G') → n_1
  42. Segment(E, E') → h_1
  43. Segment(F, F') → l_1
  44. Segment(L, A) → p_1
  45. Segment(A, A') → q_1
  46. Segment(L, B) → r_1
  47. Segment(B, B') → s_1
  48. Segment(L, C) → t_1
  49. Segment(C, C') → a_2
  50. Segment(L, D) → b_2
  51. Segment(D, D') → c_2
  52. Vector(E' + (0, 0, 0.3), E' + (0, 0, 1.3)) → u
  53. Segment(Midpoint[A', C'], Midpoint[E', G']) → d_2

AI Drawing Prompt Reference

Draw two similar cuboids, where one cuboid's edge lengths are k times those of the other. First, draw a cuboid with base ABCD and add vertex E. Then, dilate point A along line L by a factor of k to get A', and use A' as the base to draw another cuboid. Next, rotate the base quadrilateral around A' to generate the rotated faces. Finally, rotate edge AE around A' to generate the rotated edge and connect the corresponding vertices to form the new cuboid.
The prompt above was compiled from the original drawing commands and is for reference only.