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Volumes of Mouhefanggais and Spheres

Volumes of Mouhefanggais and Spheres
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2016-05-27 16:47
Solid GeometryGeometric Construction
In collections Volumes and Surface Areas of Pyramids, Cones and Spheres Volumes in Ancient Chinese Mathematics 中國古算中的體積 Volumes of Pyramid and Sphere in Ancient Chinese Mathematics Volume of Spheres in Ancient Chinese Mathematics
This example demonstrates the geometric relationship and volume calculation between Mouhefanggais and spheres. It involves core mathematical concepts such as solid geometry, volume computation, and geometric construction. The construction utilizes GeoGebra commands and objects like Point, Sequence, CircleSector, Polygon, CircleArc, Segment, Prism, OrthogonalPlane, If, Intersect, Surface, CurveCartesian, and Translate. It is suitable for high school and above, aiding in exploring solid geometry volume problems.

GeoGebra Commands & Parameters

  1. Point(Segment[I, I + (0, 0, 1)]) → P
  2. Sequence(Polygon[((-sin(θ)), (-sin(θ)), 1 - cos(θ)), (sin(θ), (-sin(θ)), 1 - cos(θ)), (sin(θ), sin(θ), 1 - cos(θ)), ((-sin(θ)), sin(θ), 1 - cos(θ))], θ, 0, π, π / n) → list4
  3. CircleSector((0, 0, 0), (0, 0, 1), (-1, 0, 0)) → d
  4. CircleSector((0, 0, 0), (0, 1, 0), (0, 0, 1)) → g
  5. Polygon((0, 0), (-1, 0), (-1, 1), (0, 1)) → poly1, i, j, k, l
  6. CircleArc((0, 0, 0), (0, 1, 0), (0, 0, 1)) → h
  7. Segment((0, 0, 0), (0, 0, 1)) → m
  8. Prism(Polygon[A, B, C, D], F) → a, G, edgeBI, edgeCJ, edgeDK, edgeFI, edgeIJ, edgeJK, edgeFK, H, E, faceABIF, faceBCJI, faceCDKJ, faceADKF, faceFIJK, edgeAF
  9. Segment(B, F) → p
  10. Segment(B, E) → q
  11. Segment(B, H) → s
  12. Segment(I, I + (0, 0, 1)) → t
  13. OrthogonalPlane(P, t) → u
  14. Prism(Polygon[(0, 0, 0), (-1, 0, 0), (-1, 1, 0), (0, 1, 0)], -1) → v, M, edge1, edge2, edge3, edge4, edge5, edge6, edge7, edge8, N, O, Q, face1, face2, face3, face4, face5
  15. If(z(P) ≠ 1, (0, sqrt(1 - z(P)^(2)), z(P)), (0, 0, 1)) → K
  16. If(z(P) ≠ 1, ((-sqrt(1 - z(P)^(2))), 0, z(P)), (0, 0, 1)) → L
  17. Intersect(q, u) → U
  18. Intersect(p, u) → V
  19. Intersect(edgeBI, u) → W
  20. Intersect(s, u) → Z
  21. Surface((s * sqrt(1 - t^(2))), sqrt(1 - t^(2)), t, s, -1, 0, t, 0, 1) → c
  22. Surface((-sqrt(1 - t^(2))), (s * sqrt(1 - t^(2))), t, s, 0, 1, t, 0, 1) → b
  23. CircleArc((0, 0, 0), (0, 0, 1), (-1, 0, 0)) → e
  24. CurveCartesian((-sqrt(1 - t^(2))), sqrt(1 - t^(2)), t, t, 0, 1) → f
  25. Intersect(edge4, u) → R
  26. Intersect(edge3, u) → S
  27. If(z(P) ≠ 1, ((-sqrt(1 - z(P)^(2))), sqrt(1 - z(P)^(2)), z(P)), (0, 0, 1)) → J
  28. Intersect(edge2, u) → T
  29. Translate(Polygon[K, R, S, T, L, J], Vector[(0, 0)]) → poly2
  30. Segment((0, 0, 0), K) → f_1
  31. Segment(A_1, K) → g_1
  32. Segment((0, 0, 0), A_1) → h_1
  33. Segment(B, W) → j_1
  34. Translate(Polygon[W, Z, U, V], Vector[(0, 0)]) → poly3
  35. Segment(W, Z) → i_1
  36. Translate(Polygon[K, R, S, T, L, J], Vector[(0, 0)]) → poly4
  37. Translate(Polygon[W, Z, U, V], Vector[(0, 0)]) → poly5
  38. Segment(G, W) → k_1
  39. Segment(A_1, (0, 0, 1)) → l_1
  40. Segment(L, A_1) → m_1
  41. Segment(V, W) → n_1
  42. Segment(A, B) → p_1
  43. Segment(B, C) → q_1
  44. Segment(C, D) → r_1
  45. Segment(D, A) → s_1
  46. Segment((0, 0), (0, 1)) → t_1
  47. Segment((0, 0), (-1, 0)) → b_1
  48. Segment((-1, 1), (0, 1)) → c_1
  49. Segment((-1, 0), (-1, 1)) → d_1