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Why Surface Area of a Closed Cylinder is 2πr² + 2πrh

Why Surface Area of a Closed Cylinder is 2πr² + 2πrh
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2020-10-25 09:15
CirclePolygonSolid GeometryTransformationSlider Animation
① This GeoGebra example dynamically demonstrates why the surface area of a closed cylinder equals the sum of two circular base areas (2πr²) and the lateral rectangular area (2πrh). ② Core mathematical concepts include surface area decomposition, area of a circle, area of a rectangle, and the relationship between 3D solids and their 2D nets. ③ Construction highlights: using GeoGebra commands such as Circle, Ellipse, Arc, Vector, Translate, Rotate, Polygon, Segment, and Angle to build the cylinder and unfold its lateral surface into a rectangle; sliders control radius r and height h to explore changes in surface area. ④ Suitable for middle and high school students learning solid geometry and surface area formulas, as well as for teachers to illustrate the derivation process in class, promoting conceptual understanding over rote memorization.

GeoGebra Commands & Parameters

  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 ≟ 0.02, B'_1, If[n < 1, B'', P]) → B_1
  65. If(n ≟ 0.02, T, If[n < 1, T', U]) → D_1
  66. If(n ≟ 0.02, H, If[n < 1, H'_1, N]) → E_1
  67. If(n ≟ 0.02, C'_1, If[n < 1, C'', Q]) → F_1
  68. Conic(B_1, D_1, E_1, F_1, I) → g_1
  69. Conic(I, B_1, D_1, E_1, F_1) → h_1
  70. If(n < 1, B, If[n < 2, B', V]) → G_1
  71. If(n < 1, A_1, If[n < 2, C_1, Z]) → H_1
  72. If(n < 1, O, If[n < 2, O', M]) → I_1
  73. If(n < 1, C, If[n < 2, C', W]) → J_1
  74. Conic(G_1, H_1, I_1, J_1, J) → k_1
  75. Vector(H, O) → i_1
  76. Translate(H, Vector[(i_1 * (n - 2))]) → H'
  77. If(n < 2, H, If[n < 3, H', O]) → K_1
  78. Segment(H, K_1) → j_1
  79. Arc(e, B, C) → q_1
  80. Intersect(g, k_1) → M_1, N_1, R_1, S_1
  81. Arc(k_1, M_1, N_1) → r_1
  82. Arc(k_1, N_1, M_1) → p_1
  83. Arc(e, C, B) → s_1
  84. OrthogonalLine(I, j) → l_1
  85. OrthogonalLine(J, l) → m_1
  86. Circle(I, w_2) → t_1
  87. Intersect(t_1, l_1, 1) → L_1
  88. Intersect(t_1, l_1, 2) → O_1
  89. OrthogonalLine(O_1, l_1) → n_1
  90. OrthogonalLine(L_1, m_1) → a_2
  91. Intersect(m_1, n_1) → P_1
  92. Intersect(m_1, a_2) → Q_1
  93. Circle(H, r) → c_2
  94. Intersect(c_2, j, 2) → T_1
  95. Circle(T_1, H) → d_2
  96. OrthogonalLine(H, j) → b_2
  97. Intersect(d_2, l_1, 1) → U_1
  98. OrthogonalLine(T_1, j) → e_2
  99. Intersect(d_2, e_2, 1) → V_1
  100. Angle(U_1, T_1, V_1) → β
  101. Circle(L_1, r) → f_2
  102. Intersect(f_2, l_1, 2) → W_1
  103. Vector(I, W_1) → u_2
  104. Translate(d_2, Vector[(u_2 * (n - 3))]) → g_2
  105. Center(g_2) → Z_1
  106. Intersect(g_2, b_2) → A_2, B_3
  107. Rotate(A_2, w_3, Z_1) → B_2
  108. If(n < 3, H, If[n < 4, B_2, L_1]) → C_2
  109. Mirror(C_2, j) → D_2
  110. Mirror(E_2, j) → F_2
  111. Midpoint(I, L_1) → G_2
  112. Intersect(m, l_1) → H_2
  113. Midpoint(H_2, C'_1) → I_2
  114. Vector(H_2, G_2) → v_2
  115. OrthogonalLine(C'_1, m) → i_2
  116. Polygon(D_2, F_2, E_2, C_2) → poly2, d_3, f_3, e_3, c_3
  117. Vector(C'_1, W_1) → z_2
  118. Translate(C'_1, Vector[(z_2 * (n - 3))]) → J_2
  119. If(n < 3, C'_1, If[n < 4, J_2, W_1]) → K_2
  120. Mirror(K_2, j) → L_2
  121. Conic(D_2, L_2, I, K_2, C_2) → h_2
  122. Arc(h_2, B_2, D_2) → k_2
  123. Arc(h_2, B_2, D_2) → p_2
  124. Conic(F_2, N_2, J, M_2, E_2) → q_2
  125. Intersect(q_2, k_1) → O_2, P_2, Q_2, R_2
  126. Center(h_2) → S_2
  127. Center(q_2) → T_2
  128. OrthogonalLine(T_2, j) → j_2
  129. OrthogonalLine(S_2, j) → m_2
  130. Intersect(h_2, m_2) → U_2, V_2
  131. Intersect(q_2, j_2) → W_2, Z_2
  132. Polygon(U_2, D_2, F_2, W_2) → poly3, u_3, d_4, f_4, w_5
  133. Polygon(V_2, C_2, E_2, Z_2) → poly4, v_3, c_4, e_4, z_3
  134. Intersect(q_2, d_3, 2) → A_3
  135. Intersect(q_2, e_3, 2) → C_3
  136. Polygon(F_2, A_3, C_3, E_2) → poly5, f_5, a_3, c_5, e_5
  137. Arc(q_2, C_3, A_3) → s_2
  138. Arc(q_2, E_2, F_2) → t_2
  139. Arc(q_2, E_2, F_2) → g_3
  140. Segment(A_3, C_3) → l_2
  141. Intersect(k_1, l_2) → D_3, E_3
  142. Segment(E_3, A_3) → n_2
  143. Segment(D_3, C_3) → b_3
  144. Segment(F_2, E_2) → h_3
  145. Arc(k_1, D_3, E_3) → k_3
  146. Segment(E_3, D_3) → i_3
  147. Segment(D_2, C_2) → j_3
  148. Segment(L_2, K_2) → l_3
  149. Arc(q_2, W_2, F_2) → p_3
  150. Arc(q_2, E_2, Z_2) → q_3
  151. Arc(q_2, E_2, Z_2) → r_3
  152. Arc(q_2, W_2, F_2) → s_3
  153. Intersect(k_1, d_3) → F_3, J_3
  154. Intersect(k_1, e_3) → G_3, K_3
  155. Arc(k_1, G_3, F_3) → t_3
  156. Arc(k_1, F_3, R_2) → g_4
  157. Arc(k_1, O_2, G_3) → h_4
  158. Segment(F_2, E_2) → r_2
  159. Intersect(k_1, r_2) → H_3, I_3
  160. Segment(F_2, I_3) → m_3
  161. Segment(E_2, H_3) → n_3
  162. Arc(q_2, E_2, F_2) → k_4
  163. Arc(k_1, H_3, I_3) → p_4
  164. Segment(I_3, H_3) → a_4
  165. Segment(F_2, E_2) → b_4
  166. Segment(E_2, Z_2) → i_4
  167. Segment(D_2, F_2) → j_4
  168. Segment(C_2, E_2) → l_4
  169. Arc(k_1, N_1, M_1) → q_4
  170. Arc(k_1, N_1, M_1) → r_4
  171. Midpoint(j_4) → L_3
  172. Translate(e, Vector[(u * h)]) → e'
  173. Arc(k_1, M_1, N_1) → s_4
  174. Arc(k_1, R_2, N_1) → t_4
  175. Arc(k_1, R_2, O_2) → d_5
  176. Arc(k_1, P_2, Q_2) → g_5
  177. Center(h_1) → M_3
  178. Segment(M_3, F_1) → a
  179. Midpoint(c_1) → N_3
  180. Point(xAxis) → O_3
  181. Circle(O_3, r) → h_5
  182. Intersect(h_5, xAxis) → P_3, Q_3
  183. Ellipse(R_3, S_3, Q_3) → k_5
  184. OrthogonalLine(S_3, i_2) → i
  185. OrthogonalLine(Q_3, i_2) → m_4
  186. OrthogonalLine(P_3, i_2) → n_4
  187. OrthogonalLine(R_3, i_2) → a_5
  188. Intersect(i_2, a_5) → T_3
  189. Intersect(n_4, i_2) → U_3
  190. Intersect(i, i_2) → V_3
  191. Intersect(m_4, i_2) → W_3
  192. Ellipse(T_3, V_3, W_3) → q_5
  193. Polygon(U_3, W_3, Q_3, P_3) → poly6, u_4, w_6, q_6, p_6
  194. Ellipse(T_3, V_3, W_3) → r_5
  195. Arc(k_5, Q_3, P_3) → s_5
  196. Arc(k_5, P_3, Q_3) → t_5
  197. Center(q_5) → Z_3
  198. Segment(O_3, Z_3) → b_5
  199. Segment(Z_3, W_3) → i_5
  200. Angle(W_3, Z_3, O_3) → γ
  201. Intersect(k_1, f_4) → A_4,
  202. Intersect(k_1, e_4) → B_4,
  203. Intersect(g_3, k_1) → C_4, D_4, E_4, F_4
  204. Polygon(Z_2, C_4, B_4) → poly7, b_6, z_4, c_6
  205. Polygon(F_4, A_4, W_2) → poly8, w_7, f_6, a_6
  206. Arc(q_2, W_2, F_4) → d_6
  207. Arc(q_2, C_4, Z_2) → e_6
  208. Arc(q_2, W_2, F_4) → g_6
  209. Arc(q_2, C_4, Z_2) → k_6
  210. Arc(k_1, G_3, J_3) → r_6
  211. Intersect(k_1, b) → G_4, H_4
  212. Segment(E_3, D_3) → m_5
  213. Segment(H_3, I_3) → n_5
  214. Point(c_3) → J_4
  215. Point(k_2) → K_4
  216. Point(c_3) → N_4
  217. Vector(M_4, N_4) → u_5
  218. Segment(D_2, C_2) → i_6
  219. Intersect(h_1, i_6) → O_4, V_4
  220. If(4 < n ≤ 5, Dilate[O_4, 5 - n, C_2]) → P_4
  221. Vector(O_4, P_4) → v_5
  222. Translate(h_1, v_5) → s_6
  223. Translate(E_1, v_5) → Q_4
  224. Center(s_6) → R_4
  225. Segment(O_4, P_4) → l_6
  226. Segment(O_4, P_4) → m_6
  227. If(4 < n ≤ 5, Rotate[Q_4, ((-(n - 4)) * 180°), R_4]) → S_4
  228. CircleArc(R_4, P_4, S_4) → t_6
  229. Mirror(O_4, j_6) → T_4
  230. Mirror(P_4, j_6) → U_4
  231. Segment(T_4, U_4) → n_6
  232. Mirror(t_6, j_6) → c_7
  233. Segment(D_2, C_2) → a_7
  234. Segment(D_2, C_2) → j_5

AI Drawing Prompt Reference

Draw a closed cylinder and its net to show surface area as two circles plus a rectangle. Start with a base circle, translate it vertically to form the top circle, and connect corresponding points with vertical segments to create the lateral surface. Then unfold the lateral surface into a rectangle whose width equals the circumference and height equals the cylinder's height. Use sliders for radius and height to dynamically adjust the shape and highlight the areas corresponding to 2πr² and 2πrh.
The prompt above was compiled from the original drawing commands and is for reference only.