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Volume of Sphere

Volume of Sphere
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2017-02-10 17:34
Solid GeometryGeometric Construction
This example demonstrates the calculation of a sphere's volume. Key mathematical concepts include the volume formula of a sphere and geometric construction. Using GeoGebra commands such as Vector, Translate, Circle, etc., the process of generating a sphere is dynamically shown. Suitable for high school and above students to learn solid geometry and volume calculation.

GeoGebra Commands & Parameters

  1. Vector(O, A) → u_1
  2. Translate(O, u_1) → O'
  3. Circle((x(A), y(A)), (R * Zoom)) → e_1
  4. Vector(B_1, Y) → v
  5. Vector(A, A1) → w_1
  6. Vector(B_1, X) → w
  7. Translate(O, Vector[(k * w)]) → C
  8. Translate(C, u_1) → C'_1
  9. Mirror(C'_1, A) → A_1
  10. Vector(A_1, C'_1) → v_1
  11. Translate(A_1, Vector[((0.5 * a) * v_1)]) → V_1
  12. Vector(B_1, Z) → u
  13. Translate(O, Vector[(R * u)]) → A_3
  14. Translate(A_3, u_1) → A'
  15. Segment(O', A') → d'
  16. Line(V_1, d') → i_1
  17. OrthogonalLine(View3D, yAxis) → a_2
  18. OrthogonalLine(View3D, xAxis) → b_2
  19. Segment(O, A_3) → d
  20. Translate(O, Vector[(R * v)]) → B
  21. Segment(O, B) → e
  22. Mirror(B, O) → D
  23. Mirror(A_3, O) → E
  24. Segment(D, O) → h
  25. Segment(E, O) → i
  26. Midpoint(B, A_3) → F
  27. Midpoint(B, F) → G
  28. Midpoint(O, G) → H
  29. Midpoint(H, G) → I
  30. Mirror(I, G) → J
  31. Conic(A_3, J, B, E, D) → f
  32. Segment(C, A_3) → j
  33. Segment(C, B) → l
  34. Segment(C, E) → m_2
  35. Segment(C, D) → n_2
  36. Line(F, O) → p
  37. Midpoint(B, E) → K
  38. Line(K, O) → q
  39. Intersect(f, p) → L, M
  40. Intersect(f, q) → N, P
  41. Segment(C, N) → r
  42. Segment(C, L) → s
  43. Segment(C, P) → t
  44. Segment(C, M) → a1
  45. Segment(C, O) → b1
  46. Translate(A, Vector[(0.8 * w)]) → V
  47. Vector(O, H1) → z
  48. Translate(f, z) → p1
  49. Translate(C, z) → I1
  50. Vector(H1, I1) → u1
  51. Translate(p1, u1) → q1
  52. Translate(D, z) → J1
  53. Translate(N, z) → K1
  54. Translate(A_3, z) → M1
  55. Translate(M, z) → N1
  56. Translate(L, z) → O1
  57. Translate(E, z) → P1
  58. Translate(P, z) → Q1
  59. Translate(B, z) → R1
  60. Translate(J1, u1) → S1
  61. Translate(K1, u1) → T1
  62. Translate(M1, u1) → U1
  63. Translate(N1, u1) → V1
  64. Translate(R1, u1) → W1
  65. Translate(Q1, u1) → Z1
  66. Translate(O1, u1) → A'2
  67. Translate(P1, u1) → B2
  68. Segment(S1, J1) → l1
  69. Segment(T1, K1) → m1
  70. Segment(U1, M1) → n1
  71. Segment(V1, N1) → r1
  72. Segment(W1, R1) → s1
  73. Segment(Z1, Q1) → t1
  74. Segment(B2, P1) → a2
  75. Segment(A'2, O1) → b2
  76. Segment(I1, H1) → g2
  77. Segment(H1, M1) → h2
  78. Mirror(V, A) → V'
  79. Translate(A, Vector[(k * w)]) → R_3
  80. Mirror(R_3, A) → R'
  81. Midpoint(R_3, A) → S
  82. Midpoint(A, R') → T
  83. Translate(A, Vector[(R * u)]) → U
  84. Mirror(U, A) → U'
  85. Translate(A, Vector[(R * v)]) → W
  86. Mirror(W, A) → W'
  87. Translate(f, u_1) → f'
  88. Midpoint(S, U') → C_1
  89. Midpoint(C_1, S) → D_1
  90. Midpoint(A, D_1) → E_1
  91. Midpoint(E_1, D_1) → F_1
  92. Mirror(F_1, D_1) → G_1
  93. Sequence(Translate[p1, Vector[(i * u1)]], i, 0, 1, o) → L1
  94. Sequence(Dilate[f, i, C], i, 0, 1, o) → L2
  95. Circle((x(A), y(A)), (R * Zoom)) → d_1
  96. Translate(C, u_1) → C'
  97. Translate(O, u_1) → O'_1
  98. Segment(C', O'_1) → b1'
  99. Intersect(d_1, e_1, 1) → H_1
  100. Intersect(e_1, b1') → I_1,
  101. Segment(I_1, A) → a_1
  102. Translate(f, u_1) → f'_1
  103. Midpoint(h2) → J_1
  104. Midpoint(d') → K_1
  105. Midpoint(d) → L_1
  106. Translate(O, u_1) → O'_3
  107. Segment(C'_1, O'_3) → b1'_1
  108. Segment(M_1, N_1) → b_1
  109. Segment(O_1, P_1) → c_1
  110. Segment(Q_1, R_1) → f_1
  111. Segment(S_1, T_1) → g_1
  112. Dilate(f, a, C) → f'_2
  113. Translate(p1, Vector[(a / 3 * u1)]) → p1'
  114. Translate(p1, Vector[(1 / 3 * u1)]) → p1'_1
  115. Translate(p1, Vector[(2 / 3 * u1)]) → p1'_2
  116. Segment(A, A1) → j_1
  117. Vector(A, A') → z_1
  118. Segment(A, W) → m_1
  119. Intersect(d_1, e_1, 1) → A_2
  120. Intersect(d_1, e_1, 1) → C_2
  121. Midpoint(C', A2) → D_2
  122. Midpoint(D_2, C') → E_2
  123. Midpoint(A, E_2) → F_2
  124. Midpoint(F_2, E_2) → G_2
  125. Midpoint(G_2, E_2) → H_2
  126. Mirror(H_2, E_2) → I_2
  127. Mirror(E_2, I_2) → J_2
  128. Conic(A2, J_2, C', W, R') → p_1
  129. Line(V_1, w_1) → n_1
  130. Line(V_1, z_1) → q_1
  131. Intersect(p_1, n_1) → W_1, Z_1
  132. Intersect(d_1, e_1, 1) → B_2
  133. Midpoint(C', U) → K_2
  134. Midpoint(K_2, C') → L_2
  135. Midpoint(A, L_2) → M_2
  136. Midpoint(M_2, L_2) → N_2
  137. Mirror(N_2, L_2) → O_2
  138. Conic(U, O_2, C', U', R') → r_1
  139. Intersect(r_1, q_1) → P_2, Q_2
  140. Midpoint(Z_1, P_2) → R_2
  141. Midpoint(Z_1, R_2) → S_2
  142. Midpoint(V_1, S_2) → T_2
  143. Midpoint(T_2, S_2) → U_2
  144. Mirror(U_2, S_2) → V_2
  145. Intersect(r_1, q_1, 2) → W_2
  146. Conic(P_2, V_2, Z_1, W_2, W_1) → s_1
  147. Sequence(Dilate[f, i, C], i, a, 1, o) → liste11
  148. Conic(P_2, V_2, Z_1, W_2, W_1) → t_1
  149. Segment(V_1, R') → h_1
  150. Sequence(Translate[q1, Vector[((-i) * u1)]], i, a / 2, 1, o) → liste12
  151. Circle((x(A), y(A)), (R * Zoom)) → k_1
  152. CircumcircleArc(P_2, R', W_2) → c_2
  153. Sequence(CurveCartesian[x(A + (((R * cos(i)) * cos(j)) * u) + (((R * cos(i)) * sin(j)) * v) + ((R * sin(i)) * w)), y(A + (((R * cos(i)) * cos(j)) * u) + (((R * cos(i)) * sin(j)) * v) + ((R * sin(i)) * w)), i, 0, 6.28319], j, 0, 6.28319, 3.141592653589793 / m) → liste15
  154. Sequence(CurveCartesian[x(A + (((R * cos(i)) * cos(j)) * u) + (((R * sin(i)) * cos(j)) * v) + ((R * sin(j)) * w)), y(A + (((R * cos(i)) * cos(j)) * u) + (((R * sin(i)) * cos(j)) * v) + ((R * sin(j)) * w)), i, 0, 6.28319], j, -1.5708, 1.5708, 3.141592653589793 / m) → liste4
  155. Sequence(CurveCartesian[x(A + (((R * cos(i)) * cos(j)) * u) + (((R * sin(i)) * cos(j)) * v) + ((R * sin(j)) * w)), y(A + (((R * cos(i)) * cos(j)) * u) + (((R * sin(i)) * cos(j)) * v) + ((R * sin(j)) * w)), i, 0, 6.28319], j, -1.5708, 1.5708, 3.141592653589793 / m) → liste1
  156. Sequence(CurveCartesian[x(A + (((R * cos(i)) * cos(j)) * u) + (((R * cos(i)) * sin(j)) * v) + ((R * sin(i)) * w)), y(A + (((R * cos(i)) * cos(j)) * u) + (((R * cos(i)) * sin(j)) * v) + ((R * sin(i)) * w)), i, 0, 6.28319], j, 0, 6.28319, 3.141592653589793 / m) → liste2
  157. Sequence((x(A), y(A)) + (R * (((cos(0°) * cos(((i * 30))°)) * u) + ((sin(0°) * cos(((i * 30))°)) * v) - (sin(((i * 30))°) * w))), i, (-n), n, dx) → list1
  158. Sequence((x(A), y(A)) + (R * (((cos(30°) * cos(((i * 30))°)) * u) + ((sin(30°) * cos(((i * 30))°)) * v) - (sin(((i * 30))°) * w))), i, (-n), n, dx) → list2
  159. Sequence((x(A), y(A)) + (R * (((cos(60°) * cos(((i * 30))°)) * u) + ((sin(60°) * cos(((i * 30))°)) * v) - (sin(((i * 30))°) * w))), i, (-n), n, dx) → list3
  160. Sequence((x(A), y(A)) + (R * (((cos(120°) * cos(((i * 30))°)) * u) + ((sin(120°) * cos(((i * 30))°)) * v) - (sin(((i * 30))°) * w))), i, (-n), n, dx) → list4
  161. Sequence((x(A), y(A)) + (R * (((cos(240°) * cos(((i * 30))°)) * u) + ((sin(240°) * cos(((i * 30))°)) * v) - (sin(((i * 30))°) * w))), i, (-n), n, dx) → list5
  162. Sequence(Conic[Element[list1, i], Element[list2, i], Element[list3, i], Element[list4, i], Element[list5, i]], i, 0, (a * 10), dx) → liste5
  163. Sequence(Dilate[f, i, C], i, b, 1, o) → liste11b