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圆柱表面积展开图

圆柱表面积展开图
几何画板及绘图内容加载中，请稍等……
2017-03-06 09:49
立体几何几何作图函数图像
①本范例演示了如何将圆柱的表面积展开为一个矩形和两个圆的平面图形；②涉及的核心数学概念包括圆柱的表面积计算、几何展开、面积公式；③作图要点——从下方『构造摘要』提炼用到了Point, Circle, Intersect, Ellipse, Vector, Translate, Polygon, Arc, Segment, Angle, Line, Midpoint, If, Conic, OrthogonalLine, Center, Rotate, Mirror等GeoGebra命令/对象；④适合高中阶段学习立体几何和表面积计算的学生使用。

GeoGebra 命令及参数

  1. Point(xAxis) → A
  2. Circle(A, r) → c
  3. Intersect(c, xAxis) → B, C
  4. Ellipse(E, D, C) → e
  5. Intersect(xAxis, yAxis) → F
  6. Vector(F, G) → u
  7. Translate(B, Vector[(u * h)]) → B'_1
  8. Translate(C, Vector[(u * h)]) → C'_1
  9. Translate(A, Vector[(u * h)]) → A'
  10. Translate(e, Vector[(u * h)]) → e'_1
  11. Polygon(B'_1, B, C, C'_1) → poly1, b', b, c_1, c'
  12. Arc(e'_1, C'_1, B'_1) → f
  13. Arc(e, B, C) → g
  14. Arc(e, C, B) → k
  15. Arc(e'_1, C'_1, B'_1) → p_5
  16. Arc(e'_1, B'_1, C'_1) → q
  17. Arc(e, B, C) → s
  18. Segment(A', A) → h_6
  19. Segment(A', C'_1) → p
  20. Angle(A, A', C'_1) → α
  21. Line(A, A') → j
  22. Intersect(e'_1, j) → H, I
  23. Intersect(k, j) → J,
  24. Circle(I, r) → t
  25. Intersect(t, j, 2) → K
  26. Circle(K, I) → d_1
  27. Circle(J, r) → e_1
  28. Intersect(e_1, j, 1) → L
  29. Circle(L, J) → f_1
  30. Intersect(f_1, j, 2) → M
  31. Intersect(d_1, j, 2) → N
  32. Intersect(s, j) → O,
  33. Vector(H, N) → v
  34. Vector(O, M) → w
  35. Translate(H, Vector[(v * n)]) → H'_1
  36. Translate(D, Vector[(u * h)]) → D'
  37. Translate(E, Vector[(u * h)]) → E'
  38. Line(B, B'_1) → l
  39. Line(C, C'_1) → m
  40. Intersect(d_1, l, 1) → P
  41. Intersect(d_1, m, 1) → Q
  42. Vector(B'_1, P) → z
  43. Vector(C'_1, Q) → u_1
  44. Translate(B'_1, Vector[(z * n)]) → B''
  45. Translate(C'_1, Vector[(u_1 * n)]) → C''
  46. Midpoint(B'_1, A') → R
  47. Midpoint(B, A) → S
  48. Line(R, S) → a_1
  49. Intersect(q, a_1) → T,
  50. Intersect(d_1, a_1, 1) → U
  51. Vector(T, U) → v_1
  52. Translate(T, Vector[(v_1 * n)]) → T'
  53. Intersect(f_1, l, 1) → V
  54. Intersect(f_1, m, 1) → W
  55. Intersect(f_1, a_1, 2) → Z
  56. Intersect(s, a_1) → A_1,
  57. Vector(B, V) → w_1
  58. Vector(A_1, Z) → z_1
  59. Vector(C, W) → b_1
  60. Translate(B, Vector[(w_1 * (n - 1))]) → B'
  61. Translate(A_1, Vector[(z_1 * (n - 1))]) → C_1
  62. Translate(O, Vector[(w * (n - 1))]) → O'
  63. Translate(C, Vector[(b_1 * (n - 1))]) → C'
  64. If(n < 1, B'', P) → B_1
  65. If(n < 1, T', U) → D_1
  66. If(n < 1, H'_1, N) → E_1
  67. If(n < 1, C'', Q) → F_1
  68. If(n < 1, B, If[n < 2, B', V]) → G_1
  69. If(n < 1, A_1, If[n < 2, C_1, Z]) → H_1
  70. If(n < 1, O, If[n < 2, O', M]) → I_1
  71. If(n < 1, C, If[n < 2, C', W]) → J_1
  72. Conic(G_1, H_1, I_1, J_1, J) → k_1
  73. Vector(H, O) → i_1
  74. Translate(H, Vector[(i_1 * (n - 2))]) → H'
  75. If(n < 2, H, If[n < 3, H', O]) → K_1
  76. Segment(H, K_1) → j_1
  77. Arc(e, B, C) → q_1
  78. Intersect(g, k_1) → M_1, N_1, R_1, S_1
  79. Arc(k_1, M_1, N_1) → r_1
  80. Arc(k_1, N_1, M_1) → p_1
  81. Arc(e, C, B) → s_1
  82. OrthogonalLine(I, j) → l_1
  83. OrthogonalLine(J, l) → m_1
  84. Circle(I, w_2) → t_1
  85. Intersect(t_1, l_1, 1) → L_1
  86. Intersect(t_1, l_1, 2) → O_1
  87. OrthogonalLine(O_1, l_1) → n_1
  88. OrthogonalLine(L_1, m_1) → a_2
  89. Intersect(m_1, n_1) → P_1
  90. Intersect(m_1, a_2) → Q_1
  91. Circle(H, r) → c_2
  92. Intersect(c_2, j, 2) → T_1
  93. Circle(T_1, H) → d_2
  94. OrthogonalLine(H, j) → b_2
  95. Intersect(d_2, l_1, 1) → U_1
  96. OrthogonalLine(T_1, j) → e_2
  97. Intersect(d_2, e_2, 1) → V_1
  98. Angle(U_1, T_1, V_1) → β
  99. Circle(L_1, r) → f_2
  100. Intersect(f_2, l_1, 2) → W_1
  101. Vector(I, W_1) → u_2
  102. Translate(d_2, Vector[(u_2 * (n - 3))]) → g_2
  103. Center(g_2) → Z_1
  104. Intersect(g_2, b_2) → A_2, B_3
  105. Rotate(A_2, w_3, Z_1) → B_2
  106. If(n < 3, H, If[n < 4, B_2, L_1]) → C_2
  107. Mirror(C_2, j) → D_2
  108. Mirror(E_2, j) → F_2
  109. Midpoint(I, L_1) → G_2
  110. Intersect(m, l_1) → H_2
  111. Midpoint(H_2, C'_1) → I_2
  112. Vector(H_2, G_2) → v_2
  113. OrthogonalLine(C'_1, m) → i_2
  114. Polygon(D_2, F_2, E_2, C_2) → poly2, d_3, f_3, e_3, c_3
  115. Vector(C'_1, W_1) → z_2
  116. Translate(C'_1, Vector[(z_2 * (n - 3))]) → J_2
  117. If(n < 3, C'_1, If[n < 4, J_2, W_1]) → K_2
  118. Mirror(K_2, j) → L_2
  119. Conic(D_2, L_2, I, K_2, C_2) → h_2
  120. Arc(h_2, B_2, D_2) → k_2
  121. Arc(h_2, B_2, D_2) → p_2
  122. Conic(F_2, N_2, J, M_2, E_2) → q_2
  123. Intersect(q_2, k_1) → O_2, P_2, Q_2, R_2
  124. Center(h_2) → S_2
  125. Center(q_2) → T_2
  126. OrthogonalLine(T_2, j) → j_2
  127. OrthogonalLine(S_2, j) → m_2
  128. Intersect(h_2, m_2) → U_2, V_2
  129. Intersect(q_2, j_2) → W_2, Z_2
  130. Polygon(U_2, D_2, F_2, W_2) → poly3, u_3, d_4, f_4, w_5
  131. Polygon(V_2, C_2, E_2, Z_2) → poly4, v_3, c_4, e_4, z_3
  132. Intersect(q_2, d_3, 2) → A_3
  133. Intersect(q_2, e_3, 2) → C_3
  134. Polygon(F_2, A_3, C_3, E_2) → poly5, f_5, a_3, c_5, e_5
  135. Arc(q_2, C_3, A_3) → s_2
  136. Arc(q_2, E_2, F_2) → t_2
  137. Arc(q_2, E_2, F_2) → g_3
  138. Segment(A_3, C_3) → l_2
  139. Intersect(k_1, l_2) → D_3, E_3
  140. Segment(E_3, A_3) → n_2
  141. Segment(D_3, C_3) → b_3
  142. Segment(F_2, E_2) → h_3
  143. Arc(k_1, D_3, E_3) → k_3
  144. Segment(E_3, D_3) → i_3
  145. Segment(D_2, C_2) → j_3
  146. Segment(L_2, K_2) → l_3
  147. Arc(q_2, W_2, F_2) → p_3
  148. Arc(q_2, E_2, Z_2) → q_3
  149. Arc(q_2, E_2, Z_2) → r_3
  150. Arc(q_2, W_2, F_2) → s_3
  151. Intersect(k_1, d_3) → F_3, J_3
  152. Intersect(k_1, e_3) → G_3, K_3
  153. Arc(k_1, G_3, F_3) → t_3
  154. Arc(k_1, F_3, R_2) → g_4
  155. Arc(k_1, O_2, G_3) → h_4
  156. Segment(F_2, E_2) → r_2
  157. Intersect(k_1, r_2) → H_3, I_3
  158. Segment(F_2, I_3) → m_3
  159. Segment(E_2, H_3) → n_3
  160. Arc(q_2, E_2, F_2) → k_4
  161. Arc(k_1, H_3, I_3) → p_4
  162. Segment(I_3, H_3) → a_4
  163. Segment(F_2, E_2) → b_4
  164. Segment(E_2, Z_2) → i_4
  165. Segment(D_2, F_2) → j_4
  166. Segment(C_2, E_2) → l_4
  167. Arc(k_1, N_1, M_1) → q_4
  168. Arc(k_1, N_1, M_1) → r_4
  169. Midpoint(j_4) → L_3
  170. Translate(e, Vector[(u * h)]) → e'
  171. Arc(k_1, M_1, N_1) → s_4
  172. Arc(k_1, R_2, N_1) → t_4
  173. Arc(k_1, R_2, O_2) → d_5
  174. Arc(k_1, P_2, Q_2) → g_5
  175. Midpoint(c_1) → N_3
  176. Point(xAxis) → O_3
  177. Circle(O_3, r) → h_5
  178. Intersect(h_5, xAxis) → P_3, Q_3
  179. Ellipse(R_3, S_3, Q_3) → k_5
  180. OrthogonalLine(S_3, i_2) → i
  181. OrthogonalLine(Q_3, i_2) → m_4
  182. OrthogonalLine(P_3, i_2) → n_4
  183. OrthogonalLine(R_3, i_2) → a_5
  184. Intersect(i_2, a_5) → T_3
  185. Intersect(n_4, i_2) → U_3
  186. Intersect(i, i_2) → V_3
  187. Intersect(m_4, i_2) → W_3
  188. Ellipse(T_3, V_3, W_3) → q_5
  189. Polygon(U_3, W_3, Q_3, P_3) → poly6, u_4, w_6, q_6, p_6
  190. Ellipse(T_3, V_3, W_3) → r_5
  191. Arc(k_5, Q_3, P_3) → s_5
  192. Arc(k_5, P_3, Q_3) → t_5
  193. Center(q_5) → Z_3
  194. Segment(O_3, Z_3) → b_5
  195. Segment(Z_3, W_3) → i_5
  196. Angle(W_3, Z_3, O_3) → γ
  197. Intersect(k_1, f_4) → A_4,
  198. Intersect(k_1, e_4) → B_4,
  199. Intersect(g_3, k_1) → C_4, D_4, E_4, F_4
  200. Polygon(Z_2, C_4, B_4) → poly7, b_6, z_4, c_6
  201. Polygon(F_4, A_4, W_2) → poly8, w_7, f_6, a_6
  202. Arc(q_2, W_2, F_4) → d_6
  203. Arc(q_2, C_4, Z_2) → e_6
  204. Arc(q_2, W_2, F_4) → g_6
  205. Arc(q_2, C_4, Z_2) → k_6
  206. Arc(k_1, G_3, J_3) → r_6
  207. Intersect(k_1, b) → G_4, H_4
  208. Segment(N_2, H_4) → j_5
  209. Segment(M_2, G_4) → l_5
  210. Segment(E_3, D_3) → m_5
  211. Segment(H_3, I_3) → n_5
  212. Point(j_4) → I_4
  213. Point(l_4) → J_4
  214. Midpoint(F_2, E_2) → K_4
  215. OrthogonalLine(K_4, f_3) → i_6
  216. Point(i_6) → L_4
  217. OrthogonalLine(L_4, i_6) → j_6
  218. Intersect(j_6, j_4) → M_4
  219. Intersect(j_6, l_4) → N_4
  220. Point(j_6) → O_4
  221. Vector(O_4, M_4) → v_4
  222. Point(j_6) → P_4
  223. Vector(P_4, N_4) → u_5

AI 绘图提示语参考

画一个圆柱体的展开图，包含一个矩形和两个圆形。矩形的一边与圆柱体的高重合，两个圆形分别代表圆柱体的上下底面。首先画出圆柱体的主视图，得到一个矩形和两个圆形的边界。然后将矩形的一边展开，使其长度等于圆柱体底面的周长，并将两个圆形分别放置在矩形的两端。最后，将展开后的图形按照实际形状绘制出来，展示圆柱体的表面如何展开成一个平面图形。
以上提示语为根据原始绘图命令整理而来，仅供参考。