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Difference of Cubes: Geometric Visualization

Difference of Cubes: Geometric Visualization
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2019-08-24 12:48
PolygonFunctionSolid GeometryTransformationGeometric Construction
This GeoGebra example demonstrates the geometric meaning of the difference of cubes formula a³ - b³ = (a-b)(a²+ab+b²) using a 3D model. It constructs two cubes of different sizes and shows that the volume remaining after subtracting the smaller cube from the larger one can be decomposed into three rectangular prisms, visually confirming the algebraic identity. Core mathematical concepts include cube volume, polynomial factorization, solid geometry volume relations, and spatial transformations such as scaling and translation. The construction uses GeoGebra commands: Point, Intersect, and Line to define key vertices and edges; Cube and Prism to build the main solids; Dilate, Vector, and Translate to position the smaller cube; and Segment to connect vertices forming faces (e.g., faceALMN, faceANRO) and edges (e.g., edgeAN, edgeAL). The final solid is composed of multiple faces (faceLMQP, faceMNRQ, faceOPQR, etc.) that enclose the remaining volume. This is suitable for high school or early college students, serving as an interactive tool for connecting algebra and geometry, and can be used by teachers for dynamic demonstrations or by students for self-exploration of factorization.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Intersect(zAxis, yAxis) → B
  3. Line(C, xAxis) → f
  4. Line(A, yAxis) → g
  5. Intersect(f, g) → F
  6. Point(Circle[L, Distance[A, L], Segment[A, L]]) → M
  7. Cube(A, L, M) → a, N, faceMNRQ, faceOPQR, edgeAN, edgeAL, edgeLM_1, edgeMN, edgeAO, edgeNR, edgeOR, edgeOP, O, edgeLP, edgePQ_1, edgeMQ, edgeQR_1, P, Q, R, faceALMN, faceANRO, faceALPO, faceLMQP
  8. Prism(Q, W, V, P, R) → b, Z, edgeQW, edgeVW, edgePV, edgePQ_2, edgeQR_2, edgeWZ, edge3, edge1, edgeRZ, edge4, A_1, edge5, edge2, B_1, facePQWV, faceQRZW, face4, face2, face1, face3
  9. Prism(D, A, S, G, K) → c, C_1, edgeAD_1, edgeAS_1, edgeGS_1, edgeDG_1, edgeDK_1, edge6, edge10, edge7, edge8, edge11, D_1, edge12, edge9, E_1, faceADGS_1, face5, face6, face8, face7, face9
  10. Prism(E, M, L, S, I) → d, F_1, edgeEM_1, edgeLM_2, edgeLS_1, edgeES_1, edgeEI_1, edge16, edge15, edge17, edge13, edge18, G_1, edge19, edge14, H_1, faceEMLS_1, face10, face13, face14, face11, face12
  11. Prism(R, B_1, P, Q, H) → e, I_1, edge25, edge22, edgePQ, edgeQR, edgeHR, edge26, edge23, edge24, edge20, edge27, J_1, edge28, edge21, K_1, face18, face16, face20, face19, face15, face17
  12. Prism(D, A, S, G, K) → h, L_1, edgeAD, edgeAS, edgeGS, edgeDG, edgeDK, edge29, edge33, edge30, edge31, edge34, M_1, edge35, edge32, N_1, faceADGS, face21, face22, face24, face23, face25
  13. Prism(E, M, L, S, I) → i, O_1, edgeEM, edgeLM, edgeLS, edgeES, edgeEI, edge39, edge38, edge40, edge36, edge41, P_1, edge42, edge37, Q_1, faceEMLS, face26, face29, face30, face27, face28
  14. Dilate(R_1, j, B) → S_1
  15. Vector(N, S_1) → u
  16. Translate(a, u) → a'
  17. Intersect(edge15, edgeLM_2) → T_1
  18. Segment(A, T_1) → k
  19. Intersect(edgeEM_1, edgeLM_2) → U_1
  20. Intersect(edgeQR_2, edgePQ_2) → V_1
  21. Segment(U_1, V_1) → l
  22. Intersect(edge2, edge1) → W_1
  23. Segment(A, W_1) → m
  24. Point(edgeEM_1) → Z_1
  25. Intersect(edgeQR_2, edgePQ_2) → A_2
  26. Segment(Z_1, A_2) → n
  27. Intersect(edge1, edge2) → B_2
  28. Segment(A, B_2) → p
  29. Intersect(edge15, edgeLM_2) → C_2
  30. Segment(A, C_2) → q

AI Drawing Prompt Reference

Create a 3D model to visualize the geometric meaning of the difference of cubes formula a³ - b³ = (a-b)(a²+ab+b²). Construct two cubes of different sizes, showing that the volume remaining after subtracting the smaller cube from the larger one can be decomposed into three rectangular prisms, visually confirming the algebraic identity. Start by defining key points and edges using the axes, then build the two cubes, position the smaller one with scaling and translation, and finally connect vertices with segments to form faces and edges, displaying the remaining volume as three prisms.
The prompt above was compiled from the original drawing commands and is for reference only.