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Maximizing Surface Area of an Open Middle Cube (V2)

Maximizing Surface Area of an Open Middle Cube (V2)
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2021-01-04 23:12
PolygonSolid GeometryOptimizationGeometric Construction
This GeoGebra example demonstrates how removing the middle portion of a cube (“open middle cube”) affects the exposed surface area, and how varying the geometric parameters can be used to study a surface-area maximization problem. It centers on solid (3D) geometry concepts such as assembling surface patches from square faces, computing/visualizing surface area contributions, and using geometric constraints that lead to linear relationships among dimensions. Construction-wise, the model builds square polygonal cross-sections with Polygon (e.g., A_2, B_2, C_2, D_2 and poly1_2, and similarly A_3, B_3, C_3, D_3 and poly1_3), then creates the 3D bodies via Prism. Explicit face objects (faceABFE_2, faceBCGF_2, faceCDHG_2, faceADHE_2, faceEFGH_2) manage which surfaces are exposed, while edge objects (edgeAE_2, edgeBF_2, edgeCG_2, edgeDH_2, edgeEF_2, edgeFG_2, edgeGH_2, edgeEH_2) track key lengths that drive the geometry. The parametrized points (a_2, b_2, c_2, d_2, e_2, and the corresponding set for the 3D construction) enable continuous exploration of how the objective changes with the configuration. Suitable for advanced high school and early university students studying optimization via geometric modeling and interactive exploration in GeoGebra.

GeoGebra Commands & Parameters

  1. Polygon(A_2, B_2, C_2, D_2) → poly1_2, a_2, b_2, c_2, d_2
  2. Prism(poly1_2, E_2) → e_2, F_2, edgeBF_2, edgeCG_2, edgeDH_2, edgeEF_2, edgeFG_2, edgeGH_2, edgeEH_2, G_2, H_2, faceABFE_2, faceBCGF_2, faceCDHG_2, faceADHE_2, faceEFGH_2, edgeAE_2
  3. Polygon(A_3, B_3, C_3, D_3) → poly1_3, a_3, b_3, c_3, d_3
  4. Prism(poly1_3, E_3) → e_3, F_3, edgeBF_3, edgeCG_3, edgeDH_3, edgeEF_3, edgeFG_3, edgeGH_3, edgeEH_3, G_3, H_3, faceABFE_3, faceBCGF_3, faceCDHG_3, faceADHE_3, faceEFGH_3, edgeAE_3
  5. Polygon(A_4, B_4, C_4, D_4) → poly1_4, a_4, b_4, c_4, d_4
  6. Prism(poly1_4, E_4) → e_4, F_4, edgeBF_4, edgeCG_4, edgeDH_4, edgeEF_4, edgeFG_4, edgeGH_4, edgeEH_4, G_4, H_4, faceABFE_4, faceBCGF_4, faceCDHG_4, faceADHE_4, faceEFGH_4, edgeAE_4
  7. Polygon(A_5, B_5, C_5, D_5) → poly1_5, a_5, b_5, c_5, d_5
  8. Prism(poly1_5, E_5) → e_5, F_5, edgeBF_5, edgeCG_5, edgeDH_5, edgeEF_5, edgeFG_5, edgeGH_5, edgeEH_5, G_5, H_5, faceABFE_5, faceBCGF_5, faceCDHG_5, faceADHE_5, faceEFGH_5, edgeAE_5
  9. Polygon(A_6, B_6, C_6, D_6) → poly1_6, a_6, b_6, c_6, d_6
  10. Prism(poly1_6, E_6) → e_6, F_6, edgeBF_6, edgeCG_6, edgeDH_6, edgeEF_6, edgeFG_6, edgeGH_6, edgeEH_6, G_6, H_6, faceABFE_6, faceBCGF_6, faceCDHG_6, faceADHE_6, faceEFGH_6, edgeAE_6
  11. Polygon(A_9, B_9, C_9, D_9) → poly1_9, a_9, b_9, c_9, d_9
  12. Prism(poly1_9, E_9) → e_9, F_9, edgeBF_9, edgeCG_9, edgeDH_9, edgeEF_9, edgeFG_9, edgeGH_9, edgeEH_9, G_9, H_9, faceABFE_9, faceBCGF_9, faceCDHG_9, faceADHE_9, faceEFGH_9, edgeAE_9
  13. Polygon(A_{10}, B_{10}, C_{10}, D_{10}) → poly1_{10}, a_{10}, b_{10}, c_{10}, d_{10}
  14. Prism(poly1_{10}, E_{10}) → e_{10}, F_{10}, edgeBF_{10}, edgeCG_{10}, edgeDH_{10}, edgeEF_{10}, edgeFG_{10}, edgeGH_{10}, edgeEH_{10}, G_{10}, H_{10}, faceABFE_{10}, faceBCGF_{10}, faceCDHG_{10}, faceADHE_{10}, faceEFGH_{10}, edgeAE_{10}
  15. Polygon(A_{11}, B_{11}, C_{11}, D_{11}) → poly1_{11}, a_{11}, b_{11}, c_{11}, d_{11}
  16. Prism(poly1_{11}, E_{11}) → e_{11}, F_{11}, edgeBF_{11}, edgeCG_{11}, edgeDH_{11}, edgeEF_{11}, edgeFG_{11}, edgeGH_{11}, edgeEH_{11}, G_{11}, H_{11}, faceABFE_{11}, faceBCGF_{11}, faceCDHG_{11}, faceADHE_{11}, faceEFGH_{11}, edgeAE_{11}
  17. Polygon(A_1, B_1, C_1, D_1) → poly1_1, a_1, b_1, c_1, d_1
  18. Prism(poly1_1, E_1) → e_1, F_1, edgeBF_1, edgeCG_1, edgeDH_1, edgeEF_1, edgeFG_1, edgeGH_1, edgeEH_1, G_1, H_1, faceABFE_1, faceBCGF_1, faceCDHG_1, faceADHE_1, faceEFGH_1, edgeAE_1
  19. Polygon(E, B, C, D) → poly1, a, b, c, d
  20. Polygon(A_{13}, B_{13}, C_{13}, D_{13}) → poly1_{13}, a_{13}, b_{13}, c_{13}, d_{13}
  21. Prism(poly1_{13}, E_{13}) → e_{13}, F_{13}, edgeBF_{13}, edgeCG_{13}, edgeDH_{13}, edgeEF_{13}, edgeFG_{13}, edgeGH_{13}, edgeEH_{13}, G_{13}, H_{13}, faceABFE_{13}, faceBCGF_{13}, faceCDHG_{13}, faceADHE_{13}, faceEFGH_{13}, edgeAE_{13}
  22. Polygon(A_{14}, B_{14}, C_{14}, D_{14}) → poly1_{14}, a_{14}, b_{14}, c_{14}, d_{14}
  23. Prism(poly1_{14}, E_{14}) → e_{14}, F_{14}, edgeBF_{14}, edgeCG_{14}, edgeDH_{14}, edgeEF_{14}, edgeFG_{14}, edgeGH_{14}, edgeEH_{14}, G_{14}, H_{14}, faceABFE_{14}, faceBCGF_{14}, faceCDHG_{14}, faceADHE_{14}, faceEFGH_{14}, edgeAE_{14}
  24. Polygon(A_{15}, B_{15}, C_{15}, D_{15}) → poly1_{15}, a_{15}, b_{15}, c_{15}, d_{15}
  25. Prism(poly1_{15}, E_{15}) → e_{15}, F_{15}, edgeBF_{15}, edgeCG_{15}, edgeDH_{15}, edgeEF_{15}, edgeFG_{15}, edgeGH_{15}, edgeEH_{15}, G_{15}, H_{15}, faceABFE_{15}, faceBCGF_{15}, faceCDHG_{15}, faceADHE_{15}, faceEFGH_{15}, edgeAE_{15}
  26. Polygon(A_{16}, B_{16}, C_{16}, D_{16}) → poly1_{16}, a_{16}, b_{16}, c_{16}, d_{16}
  27. Prism(poly1_{16}, E_{16}) → e_{16}, F_{16}, edgeBF_{16}, edgeCG_{16}, edgeDH_{16}, edgeEF_{16}, edgeFG_{16}, edgeGH_{16}, edgeEH_{16}, G_{16}, H_{16}, faceABFE_{16}, faceBCGF_{16}, faceCDHG_{16}, faceADHE_{16}, faceEFGH_{16}, edgeAE_{16}
  28. Polygon(A_{17}, B_{17}, C_{17}, D_{17}) → poly1_{17}, a_{17}, b_{17}, c_{17}, d_{17}
  29. Prism(poly1_{17}, E_{17}) → e_{17}, F_{17}, edgeBF_{17}, edgeCG_{17}, edgeDH_{17}, edgeEF_{17}, edgeFG_{17}, edgeGH_{17}, edgeEH_{17}, G_{17}, H_{17}, faceABFE_{17}, faceBCGF_{17}, faceCDHG_{17}, faceADHE_{17}, faceEFGH_{17}, edgeAE_{17}
  30. Polygon(A_{18}, B_{18}, C_{18}, D_{18}) → poly1_{18}, a_{18}, b_{18}, c_{18}, d_{18}
  31. Prism(poly1_{18}, E_{18}) → e_{18}, F_{18}, edgeBF_{18}, edgeCG_{18}, edgeDH_{18}, edgeEF_{18}, edgeFG_{18}, edgeGH_{18}, edgeEH_{18}, G_{18}, H_{18}, faceABFE_{18}, faceBCGF_{18}, faceCDHG_{18}, faceADHE_{18}, faceEFGH_{18}, edgeAE_{18}
  32. Polygon(A_{19}, B_{19}, C_{19}, D_{19}) → poly1_{19}, a_{19}, b_{19}, c_{19}, d_{19}
  33. Prism(poly1_{19}, E_{19}) → e_{19}, F_{19}, edgeBF_{19}, edgeCG_{19}, edgeDH_{19}, edgeEF_{19}, edgeFG_{19}, edgeGH_{19}, edgeEH_{19}, G_{19}, H_{19}, faceABFE_{19}, faceBCGF_{19}, faceCDHG_{19}, faceADHE_{19}, faceEFGH_{19}, edgeAE_{19}
  34. Polygon(A_{21}, B_{21}, C_{21}, D_{21}) → poly1_{21}, a_{21}, b_{21}, c_{21}, d_{21}
  35. Prism(poly1_{21}, E_{21}) → e_{21}, F_{21}, edgeBF_{21}, edgeCG_{21}, edgeDH_{21}, edgeEF_{21}, edgeFG_{21}, edgeGH_{21}, edgeEH_{21}, G_{21}, H_{21}, faceABFE_{21}, faceBCGF_{21}, faceCDHG_{21}, faceADHE_{21}, faceEFGH_{21}, edgeAE_{21}
  36. Polygon(A_{22}, B_{22}, C_{22}, D_{22}) → poly1_{22}, a_{22}, b_{22}, c_{22}, d_{22}
  37. Prism(poly1_{22}, E_{22}) → e_{22}, F_{22}, edgeBF_{22}, edgeCG_{22}, edgeDH_{22}, edgeEF_{22}, edgeFG_{22}, edgeGH_{22}, edgeEH_{22}, G_{22}, H_{22}, faceABFE_{22}, faceBCGF_{22}, faceCDHG_{22}, faceADHE_{22}, faceEFGH_{22}, edgeAE_{22}
  38. Polygon(A_{24}, B_{24}, C_{24}, D_{24}) → poly1_{24}, a_{24}, b_{24}, c_{24}, d_{24}
  39. Prism(poly1_{24}, E_{24}) → e_{24}, F_{24}, edgeBF_{24}, edgeCG_{24}, edgeDH_{24}, edgeEF_{24}, edgeFG_{24}, edgeGH_{24}, edgeEH_{24}, G_{24}, H_{24}, faceABFE_{24}, faceBCGF_{24}, faceCDHG_{24}, faceADHE_{24}, faceEFGH_{24}, edgeAE_{24}
  40. Polygon(E_{25}, B_{25}, C_{25}, D_{25}) → poly1_{25}, a_{25}, b_{25}, c_{25}, d_{25}
  41. Polygon(A_{25}, B_{26}, C_{26}, D_{26}) → poly1_{26}, a_{26}, b_{26}, c_{26}, d_{26}
  42. Prism(poly1_{26}, E_{26}) → e_{25}, F_{25}, edgeBF_{25}, edgeCG_{25}, edgeDH_{25}, edgeEF_{25}, edgeFG_{25}, edgeGH_{25}, edgeEH_{25}, G_{25}, H_{25}, faceABFE_{25}, faceBCGF_{25}, faceCDHG_{25}, faceADHE_{25}, faceEFGH_{25}, edgeAE_{25}
  43. Polygon(A_{26}, B_{27}, C_{27}, D_{27}) → poly1_{27}, a_{27}, b_{27}, c_{27}, d_{27}
  44. Prism(poly1_{27}, E_{27}) → e_{26}, F_{26}, edgeBF_{26}, edgeCG_{26}, edgeDH_{26}, edgeEF_{26}, edgeFG_{26}, edgeGH_{26}, edgeEH_{26}, G_{26}, H_{26}, faceABFE_{26}, faceBCGF_{26}, faceCDHG_{26}, faceADHE_{26}, faceEFGH_{26}, edgeAE_{26}
  45. Polygon(A_{27}, B_{28}, C_{28}, D_{28}) → poly1_{28}, a_{28}, b_{28}, c_{28}, d_{28}
  46. Prism(poly1_{28}, E_{28}) → e_{27}, F_{27}, edgeBF_{27}, edgeCG_{27}, edgeDH_{27}, edgeEF_{27}, edgeFG_{27}, edgeGH_{27}, edgeEH_{27}, G_{27}, H_{27}, faceABFE_{27}, faceBCGF_{27}, faceCDHG_{27}, faceADHE_{27}, faceEFGH_{27}, edgeAE_{27}
  47. Polygon(A_{28}, B_{29}, C_{29}, D_{29}) → poly1_{29}, a_{29}, b_{29}, c_{29}, d_{29}
  48. Prism(poly1_{29}, E_{29}) → e_{28}, F_{28}, edgeBF_{28}, edgeCG_{28}, edgeDH_{28}, edgeEF_{28}, edgeFG_{28}, edgeGH_{28}, edgeEH_{28}, G_{28}, H_{28}, faceABFE_{28}, faceBCGF_{28}, faceCDHG_{28}, faceADHE_{28}, faceEFGH_{28}, edgeAE_{28}
  49. Polygon(A_{29}, B_{30}, C_{30}, D_{30}) → poly1_{30}, a_{30}, b_{30}, c_{30}, d_{30}
  50. Prism(poly1_{30}, E_{30}) → e_{29}, F_{29}, edgeBF_{29}, edgeCG_{29}, edgeDH_{29}, edgeEF_{29}, edgeFG_{29}, edgeGH_{29}, edgeEH_{29}, G_{29}, H_{29}, faceABFE_{29}, faceBCGF_{29}, faceCDHG_{29}, faceADHE_{29}, faceEFGH_{29}, edgeAE_{29}
  51. Polygon(A_{30}, B_{31}, C_{31}, D_{31}) → poly1_{31}, a_{31}, b_{31}, c_{31}, d_{31}
  52. Prism(poly1_{31}, E_{31}) → e_{30}, F_{30}, edgeBF_{30}, edgeCG_{30}, edgeDH_{30}, edgeEF_{30}, edgeFG_{30}, edgeGH_{30}, edgeEH_{30}, G_{30}, H_{30}, faceABFE_{30}, faceBCGF_{30}, faceCDHG_{30}, faceADHE_{30}, faceEFGH_{30}, edgeAE_{30}
  53. Polygon(A_{31}, B_{32}, C_{32}, D_{32}) → poly1_{32}, a_{32}, b_{32}, c_{32}, d_{32}
  54. Prism(poly1_{32}, E_{32}) → e_{31}, F_{31}, edgeBF_{31}, edgeCG_{31}, edgeDH_{31}, edgeEF_{31}, edgeFG_{31}, edgeGH_{31}, edgeEH_{31}, G_{31}, H_{31}, faceABFE_{31}, faceBCGF_{31}, faceCDHG_{31}, faceADHE_{31}, faceEFGH_{31}, edgeAE_{31}
  55. Polygon(A_{32}, B_{33}, C_{33}, D_{33}) → poly1_{33}, a_{33}, b_{33}, c_{33}, d_{33}
  56. Prism(poly1_{33}, E_{33}) → e_{32}, F_{32}, edgeBF_{32}, edgeCG_{32}, edgeDH_{32}, edgeEF_{32}, edgeFG_{32}, edgeGH_{32}, edgeEH_{32}, G_{32}, H_{32}, faceABFE_{32}, faceBCGF_{32}, faceCDHG_{32}, faceADHE_{32}, faceEFGH_{32}, edgeAE_{32}
  57. Polygon(E_{38}, B_{38}, C_{38}, D_{38}) → poly1_{38}, a_{38}, b_{38}, c_{38}, d_{38}

AI Drawing Prompt Reference

Create a 3D “open middle cube” geometric setup for a surface-area maximization study. First draw several square cross-sections in the same orientation, defining their four corner points and keeping them as parameter-driven square shapes. Then extrude each square along a vertical direction to form a prism so that corresponding top and bottom points connect, producing a cube-like solid. Explicitly distinguish which side faces remain exposed as outer patches (the rectangular faces assembled around the opening) while the central removed portion is left open. Track the key edge lengths along the important connecting edges, so the exposed surface geometry varies with the chosen dimensions. Optionally build multiple similar configurations by changing the parameters to compare surface-area contributions.
The prompt above was compiled from the original drawing commands and is for reference only.