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Net of a Cube: Exploring Surface Area

Net of a Cube: Exploring Surface Area
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2018-05-01 19:17
PolygonSolid GeometryTransformationGeometric ConstructionSlider Animation
In collections Surface Area Discovery
① This interactive applet demonstrates the net of a cube, allowing learners to visually explore how a three-dimensional cube unfolds into a two-dimensional pattern and verify the surface area formula. ② Core mathematical concepts include solid geometry, cube nets, surface area calculation, and spatial visualization skills. ③ Construction highlights: Based on the construction summary, the applet uses GeoGebra commands such as Point, Segment, Polygon, Circle, Line, Intersect, Rotate, OrthogonalLine, CircleArc, Vector, Translate, Ray, LineBisector, If, and Midpoint, creating multiple polygons (poly1 to poly4) and auxiliary lines, rotations, and translations to animate the unfolding and folding of the cube's six faces. ④ Suitable for upper elementary to middle school students learning introductory solid geometry, as well as for teachers demonstrating surface area derivation in class or for self-directed exploration.

GeoGebra Commands & Parameters

  1. Point(Circle[A, s]) → B
  2. Segment(A, B) → a
  3. Polygon(A, B, 4) → poly1, a_1, b, c, d, C, D
  4. Circle(B, e) → f
  5. Line(A, B) → g
  6. Intersect(f, g, 2) → E
  7. Rotate(E, 30°, B) → E'
  8. OrthogonalLine(E', g) → h
  9. Segment(B, E') → i
  10. Line(C, i) → j
  11. Intersect(h, j) → F
  12. Polygon(E', F, 4) → poly2, e', f_1, k, l, G, H
  13. Polygon(A, H, G, D) → poly3, a_2, h_1, g_1, d_1
  14. Polygon(C, F, E', B) → poly4, c_1, f_2, e'_1, b_1
  15. Circle(B, A) → p
  16. Circle(A, B) → q
  17. Intersect(q, g, 2) → I
  18. Intersect(p, g, 2) → J
  19. Circle(J, B) → r
  20. Intersect(r, g, 2) → K
  21. Line(J, e'_1) → m
  22. Line(K, e'_1) → n
  23. Line(I, a_2) → t
  24. Line(H, E') → e_1
  25. Intersect(t, e_1) → L
  26. Intersect(m, e_1) → M
  27. Intersect(n, e_1) → N
  28. Line(B, E') → i_1
  29. Line(A, H) → j_1
  30. Intersect(f, i_1, 2) → O
  31. Line(O, g) → k_1
  32. Circle(E', B) → p_1
  33. Intersect(p_1, i_1, 2) → P
  34. Line(P, e_1) → l_1
  35. Intersect(j_1, l_1) → Q
  36. Line(C, B) → m_1
  37. Circle(C, B) → q_1
  38. Intersect(q_1, m_1, 2) → R
  39. Line(R, c_1) → n_1
  40. Intersect(h, n_1) → S
  41. Intersect(j_1, k_1) → T
  42. CircleArc(C, R, D) → r_1
  43. CircleArc(B, K, R) → s_1
  44. CircleArc(E', N, S) → t_1
  45. CircleArc(E', M, F) → c_2
  46. CircleArc(B, J, C) → d_2
  47. CircleArc(A, B, I) → e_2
  48. CircleArc(H, G, L) → g_2
  49. CircleArc(F, S, G) → h_2
  50. Vector(F, P) → u
  51. Vector(C, O) → v
  52. Vector(D, R) → w
  53. Vector(S, N) → u_1
  54. Vector(D, I) → v_1
  55. Translate(D, Vector[(o * w)]) → D'_1
  56. Ray(C, D'_1) → b_2
  57. Intersect(r_1, b_2, 1) → U
  58. Translate(G, Vector[(o * w)]) → G'_1
  59. Ray(F, G'_1) → i_2
  60. Intersect(h_2, i_2, 1) → V
  61. LineBisector(F, P) → j_2
  62. Intersect(f, j_2, 2) → W
  63. CircleArc(W, P, F) → k_2
  64. Line(W, l_1) → l_2
  65. LineBisector(G, Q) → m_2
  66. Intersect(l_2, m_2) → Z
  67. CircleArc(Z, Q, G) → p_2
  68. Translate(C, Vector[((o - 1) * v)]) → C'
  69. Translate(D, Vector[((o - 1) * v)]) → D'_2
  70. If(o < 1, C, If[o < 2, C', O]) → A_1
  71. If(o < 1, D, If[o < 2, D'_2, T]) → B_1
  72. Translate(D, Vector[((o - 2) * v_1)]) → D'
  73. Translate(G, Vector[((o - 2) * v_1)]) → G'_2
  74. Ray(A, D') → n_2
  75. Ray(H, G'_2) → q_2
  76. Intersect(e_2, n_2, 1) → C_1
  77. Intersect(g_2, q_2) → D_1,
  78. If(o < 2, G, If[o < 3, D_1, L]) → E_1
  79. If(o < 2, D, If[o < 3, C_1, I]) → F_1
  80. Translate(G, Vector[((o - 3) * u)]) → G'
  81. Translate(F, Vector[((o - 3) * u)]) → F'
  82. Ray(H, G') → r_2
  83. Ray(E', F') → s_2
  84. Intersect(p_2, r_2, 1) → G_1
  85. Intersect(k_2, s_2, 1) → H_1
  86. If(o < 3, G, If[o < 4, G_1, Q]) → I_1
  87. If(o < 3, F, If[o < 4, H_1, P]) → J_1
  88. Translate(S, Vector[((o - 4) * u_1)]) → S'
  89. Ray(E', S') → t_2
  90. Intersect(t_1, t_2, 1) → K_1
  91. Intersect(c_2, t_2, 1) → L_1
  92. Translate(R, Vector[((o - 4) * u_1)]) → R'
  93. Ray(B, R') → a_3
  94. Intersect(s_1, a_3, 1) → M_1
  95. Intersect(d_2, a_3, 1) → N_1
  96. If(o < 1, U, If[o < 4, R, If[o < 5, M_1, K]]) → O_1
  97. If(o < 1, V, If[o < 4, S, If[o < 5, K_1, N]]) → P_1
  98. If(o < 4, C, If[o < 5, N_1, J]) → Q_1
  99. If(o < 4, F, If[o < 5, L_1, M]) → R_1
  100. Midpoint(B, E') → S_1
  101. Midpoint(A, H) → T_1
  102. Line(T_1, S_1) → c_3
  103. Circle(S_1, a / 2) → d_3
  104. OrthogonalLine(S_1, c_3) → g_3
  105. Intersect(d_3, g_3) → U_1, V_1
  106. Polygon(U_1, V_1, 4) → poly11, u_2, v_2, k_3, l_3, W_1, Z_1
  107. Polygon(Z_1, W_1, 4) → poly12, z_1, w_1, m_3, n_3, A_2, B_2
  108. Polygon(U_1, Z_1, 4) → poly13, u_3, z_2, s_3, t_3, C_2, D_2
  109. Polygon(W_1, V_1, 4) → poly14, w_2, v_3, c_4, d_4, E_2, F_2
  110. Polygon(V_1, U_1, 4) → poly15, v_4, u_4, e_4, f_4, G_2, H_2
  111. Polygon(H_2, G_2, 4) → poly16, h_6, g_4, i_4, j_4, I_2, J_2
  112. Segment(B, U_1) → m_4
  113. LineBisector(m_4) → n_4
  114. Vector(B, U_1) → z
  115. Vector(T, C_2) → z_4
  116. Vector(L, A_2) → u_5
  117. Vector(P, E_2) → w_5
  118. Translate(I, Vector[(z * (o - 5))]) → I'
  119. Translate(A, Vector[(z * (o - 5))]) → A'
  120. Translate(B, Vector[(z * (o - 5))]) → B'
  121. Translate(J, Vector[(z * (o - 5))]) → J'
  122. Translate(K, Vector[(z * (o - 5))]) → K'
  123. Translate(L, Vector[(u_5 * (o - 5))]) → L'
  124. Translate(H, Vector[(u_5 * (o - 5))]) → H'
  125. Translate(E', Vector[(u_5 * (o - 5))]) → E''
  126. Translate(M, Vector[(u_5 * (o - 5))]) → M'
  127. Translate(N, Vector[(u_5 * (o - 5))]) → N'
  128. Translate(T, Vector[(z_4 * (o - 5))]) → T'
  129. Translate(O, Vector[(z_4 * (o - 5))]) → O'
  130. Translate(P, Vector[(w_5 * (o - 5))]) → P'
  131. Translate(Q, Vector[(w_5 * (o - 5))]) → Q'
  132. If(o < 5, O_1, If[o < 6, K', I_2]) → K_2
  133. If(o < 5, P_1, If[o < 6, N', J_2]) → L_2
  134. If(o < 5, Q_1, If[o < 6, J', G_2]) → M_2
  135. If(o < 5, R_1, If[o < 6, M', H_2]) → N_2
  136. If(o < 5, E', If[o < 6, E'', V_1]) → O_2
  137. If(o < 5, H, If[o < 6, H', W_1]) → P_2
  138. If(o < 5, E_1, If[o < 6, L', A_2]) → Q_2
  139. If(o < 5, J_1, If[o < 6, P', E_2]) → R_2
  140. If(o < 5, I_1, If[o < 6, Q', F_2]) → S_2
  141. If(o < 5, B, If[o < 6, B', U_1]) → T_2
  142. If(o < 5, A, If[o < 6, A', Z_1]) → U_2
  143. If(o < 5, A_1, If[o < 6, O', D_2]) → V_2
  144. If(o < 5, B_1, If[o < 6, T', C_2]) → W_2
  145. If(o < 5, F_1, If[o < 6, I', B_2]) → Z_2
  146. Polygon(Z_2, Q_2, P_2, U_2) → poly5, z_3, q_3, p_3, u_6
  147. Polygon(U_2, T_2, V_2, W_2) → poly6, u_7, t_4, v_5, w_3
  148. Polygon(U_2, P_2, O_2, T_2) → poly7, u_8, p_4, o_2, t_5
  149. Polygon(O_2, N_2, M_2, T_2) → poly8, o_1, n_5, m_5, t_6
  150. Polygon(N_2, L_2, K_2, M_2) → poly9, n_6, l_4, k_4, m_6
  151. Polygon(O_2, R_2, S_2, P_2) → poly10, o_3, r_3, s_4, p_5
  152. Intersect(k, v_5) → B_3
  153. Segment(G, B_3) → b_3
  154. Intersect(v_5, a_2) → A_3
  155. Segment(P_2, A_3) → e_3
  156. Intersect(t_4, p_5) → C_3
  157. Intersect(m_5, p_5) → D_3
  158. Segment(D_3, C_3) → f_3
  159. Segment(D_3, P_2) → h_3
  160. Point(Circle[E_3, s]) → F_3
  161. Segment(E_3, F_3) → j_3
  162. Line(E_3, F_3) → a_4
  163. Circle(F_3, e) → h_4
  164. Intersect(h_4, a_4, 2) → G_3
  165. Rotate(G_3, 30°, F_3) → H_3
  166. Segment(F_3, H_3) → b_4
  167. Line(H_3, j_3) → q_4
  168. Line(E_3, b_4) → r_4
  169. Intersect(q_4, r_4) → I_3
  170. OrthogonalLine(H_3, q_4) → a_5
  171. OrthogonalLine(F_3, j_3) → b_5
  172. OrthogonalLine(E_3, a_4) → c_5
  173. OrthogonalLine(I_3, q_4) → d_5
  174. Circle(E_3, s) → e_5
  175. Intersect(e_5, c_5, 2) → J_3
  176. OrthogonalLine(J_3, c_5) → f_5
  177. Circle(I_3, s) → g_5
  178. Intersect(g_5, d_5, 2) → K_3
  179. OrthogonalLine(K_3, d_5) → h_5
  180. Intersect(a_5, h_5) → L_3
  181. Intersect(b_5, f_5) → M_3
  182. Polygon(J_3, M_3, L_3, K_3) → poly17, j_5, m_7, l_5, k_5
  183. Polygon(J_3, E_3, F_3, M_3) → poly18, j_6, e_6, f_6, m_8
  184. Polygon(F_3, M_3, L_3, H_3) → poly19, f_7, m_9, l_6, h_7
  185. Polygon(I_3, K_3, L_3, H_3) → poly20, i_5, k_6, l_7, h_8
  186. Polygon(E_3, J_3, K_3, I_3) → poly21, e_7, j_7, k_7, i_6
  187. Polygon(E_3, I_3, H_3, F_3) → poly22, e_8, i_7, h_9, f_8

AI Drawing Prompt Reference

Draw the net of a cube, showing how its six faces unfold into a 2D pattern. Start with a square, then rotate and draw adjacent squares around one edge to form a foldable layout. Use auxiliary lines, arcs, and rotation animations to demonstrate how the cube folds and unfolds, labeling key points and lines for surface area calculation understanding.
The prompt above was compiled from the original drawing commands and is for reference only.