计算复合固体体积：探索矩形棱柱组合的体积公式

GeoGebra 命令及参数

1. Point(Segment[A_{1} + (2, 0, 0), A_{1} + (8, 0, 0)]) → ED_{1}
2. Sequence(Translate[A_{1}, Vector[(ii, 0, 0)]], ii, 1, x(ED_{1}) + 4 - 1) → l5
3. Point(Segment[ED_{1} + (0, 1, 0), ED_{1} + (0, 8, 0)]) → FD_{1}
4. Point(l5) → CD_{1}
5. Point(Segment[A_{1} + (0, 0, 1), A_{1} + (0, 0, 8)]) → GD_{1}
6. Zip((x(CD_{1}), y(CD_{1}), kk), kk, Sequence[1, 8]) → l4
7. Point(l4) → ID_{1}
8. Polygon({A_{1}, B_{1}, DD_{1}, CD_{1}}) → base1_{1}
9. Polygon({CD_{1}, ED_{1}, FD_{1}, DD_{1}}) → base2_{1}
10. Prism(base1_{1}, GD_{1}) → prism1, K_{1}, edge2_{1}, edge3_{1}, edge4_{1}, edgeGDK_{1}, edgeKL_{1}, edgeLM_{1}, edgeGDM_{1}, L_{1}, M_{1}, face1_{1}, face3_{1}, face4_{1}, face2_{1}, faceGDKLM_{1}, edge1_{1}
11. Prism(base2_{1}, ID_{1}) → prism2, N_{1}, edge6_{1}, edge7_{1}, edge8_{1}, edgeIDN_{1}, edgeNO_{1}, edgeOP_{1}, edgeIDP_{1}, O_{1}, P_{1}, face5_{1}, face7_{1}, face8_{1}, face6_{1}, faceIDNOP_{1}, edge5_{1}
12. Polygon({A_{1}, ED_{1}, FD_{1}, B_{1}}) → base3_{1}
13. Prism(base3_{1}, If[z(GD_{1}) ≤ z(ID_{1}), z(GD_{1}), z(ID_{1})]) → prism3, Q_{1}, edge9_{1}, edge10_{1}, edge11_{1}, edge12_{1}, edgeQR_{1}, edgeRS_{1}, edgeST_{1}, edgeQT_{1}, R_{1}, S_{1}, T_{1}, face9_{1}, face11_{1}, face12_{1}, face10_{1}, faceQRST_{1}
14. If(z(ID_{1}) ≥ z(GD_{1}), Prism[Translate[Polygon[{CD_{1}, ED_{1}, FD_{1}, DD_{1}}], Vector[A_{1}, GD_{1}]], ID_{1}], Prism[Translate[Polygon[{A_{1}, CD_{1}, DD_{1}, B_{1}}], Vector[CD_{1}, ID_{1}]], GD_{1}]) → prism4
15. Sequence(Translate[A_{1}, Vector[(jj, 0, 0)]], jj, 0, x(CD_{1}) - x(A_{1})) → l1
16. Translate(l1, Vector[A_{1}, B_{1}]) → l1'
17. Translate(l1, Vector[A_{1}, GD_{1}]) → l1'_{1}
18. Zip(Segment[AA, BB], AA, l1, BB, l1') → squaresBase1
19. Translate(squaresBase1, Vector[A_{1}, GD_{1}]) → squaresBase1'
20. Zip(Segment[AA, BB], AA, l1, BB, l1'_{1}) → squaresFront1
21. Translate(squaresFront1, Vector[A_{1}, B_{1}]) → squaresFront1'
22. Sequence(Translate[A_{1}, Vector[(0, jj, 0)]], jj, 0, y(B_{1}) - y(A_{1})) → l2
23. Translate(l2, Vector[A_{1}, GD_{1}]) → l2'
24. Sequence(Translate[A_{1}, Vector[(0, 0, jj)]], jj, 0, z(GD_{1}) - z(A_{1})) → l3
25. Translate(l3, Vector[A_{1}, B_{1}]) → l3'
26. Zip(Segment[AA, BB], AA, l3, BB, l3') → squaresLeft1
27. Zip(Segment[AA, BB], AA, l2, BB, l2') → squaresLeft2
28. Translate(squaresLeft1, Vector[A_{1}, CD_{1}]) → squaresLeft1'
29. Translate(squaresLeft2, Vector[A_{1}, CD_{1}]) → squaresLeft2'
30. Translate(l3, Vector[A_{1}, CD_{1}]) → l3'_{1}
31. Zip(Segment[AA, BB], AA, l3, BB, l3'_{1}) → squaresBack
32. Translate(squaresBack, Vector[A_{1}, B_{1}]) → squaresBack'
33. Translate(l2', Vector[A_{1}, CD_{1}]) → l2''
34. Zip(Segment[AA, BB], AA, l2', BB, l2'') → squaresBottom
35. Translate(squaresBottom, Vector[GD_{1}, A_{1}]) → squaresBottom'
36. Sequence(Sequence[Polygon[CD_{1} + (ii, 0, jj), CD_{1} + (ii + 1, 0, jj), CD_{1} + (ii + 1, 0, jj + 1), CD_{1} + (ii, 0, jj + 1)], ii, 0, x(ED_{1}) - x(CD_{1}) - 1], jj, 0, z(N_{1}) - 1) → squaresFront2, {{1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}}
37. Sequence(Sequence[Polygon[ED_{1} + (0, ii, jj), ED_{1} + (0, ii + 1, jj), ED_{1} + (0, ii + 1, jj + 1), ED_{1} + (0, ii, jj + 1)], ii, 0, y(FD_{1}) - y(ED_{1}) - 1], jj, 0, z(N_{1}) - 1) → squaresRight, {{1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}}
38. Translate(squaresFront2, Vector[ED_{1}, FD_{1}]) → squaresFront2'
39. Translate(squaresRight, Vector[ED_{1}, CD_{1}]) → squaresRight'
40. Sequence(Sequence[Polygon[CD_{1} + (ii, jj, 0), CD_{1} + (ii + 1, jj, 0), CD_{1} + (ii + 1, jj + 1, 0), CD_{1} + (ii, jj + 1, 0)], ii, 0, x(ED_{1}) - x(CD_{1}) - 1], jj, 0, y(FD_{1}) - y(ED_{1}) - 1) → squaresBottom2, {{1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}}
41. Translate(squaresBottom2, Vector[ED_{1}, N_{1}]) → squaresBottom2'
42. If(z(GD_{1}) ≥ z(ID_{1}), abs(x(GD_{1}) - x(ID_{1})), abs(x(ID_{1}) - x(ED_{1}))) → base4
43. Corner(5) → A
44. Min(z(GD_{1}), z(ID_{1})) → hei3