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Net of a Square Pyramid (Formula Hidden)

Net of a Square Pyramid (Formula Hidden)
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2020-02-07 07:31
PolygonVectorSolid GeometryGeometric Construction
This example demonstrates the net of a square pyramid, visually representing how its surface consists of a base and four triangular faces. Core math concepts include polygons, solid geometry, angle calculations, and vector operations. GeoGebra commands like Point, Line, Circle, Rotate, Polygon, Vector, and Translate are used in construction. Suitable for high school mathematics education, helping students understand the unfolding and reconstruction of geometric solids and enhancing spatial reasoning skills.

GeoGebra Commands & Parameters

  1. Point(Circle[A, w]) → B
  2. Segment(A, B) → a
  3. Line(A, B) → b_2
  4. Circle(B, (3 * w / 4.5)) → c
  5. Intersect(c, b_2, 2) → C
  6. Rotate(C, 30°, B) → C'
  7. Line(C', b_2) → d
  8. Segment(B, C') → e
  9. Line(A, e) → f
  10. Intersect(d, f) → D
  11. Polygon(A, B, C', D) → poly1, a_1, b_1, c', d_1
  12. Segment(A, C') → g
  13. Segment(D, B) → h_1
  14. Intersect(g, h_1) → E
  15. OrthogonalLine(E, a_1) → i
  16. Circle(E, h) → k
  17. Intersect(k, i, 2) → F
  18. Segment(B, F) → j
  19. Segment(C', F) → l_5
  20. Segment(F, D) → m
  21. Segment(F, A) → n
  22. Segment(E, F) → p
  23. Segment(A, D) → q
  24. Midpoint(b_1) → G
  25. Segment(G, F) → r
  26. Angle(G, E, F) → α
  27. Segment(E, G) → t
  28. Line(E, G) → c_1
  29. Circle(G, F) → f_1
  30. Intersect(f_1, c_1, 2) → H
  31. Intersect(c_1, d_1) → I
  32. Circle(I, F) → g_1
  33. Intersect(g_1, c_1, 1) → J
  34. Vector(F, J) → u
  35. Point(c') → L
  36. Circle(L, F) → q_1
  37. Vector(F, H) → w_3
  38. Midpoint(A, B) → K
  39. Line(K, E) → e_1
  40. Circle(K, h) → p_1
  41. Intersect(e_1, c') → M
  42. Circle(M, h) → r_1
  43. Intersect(r_1, e_1, 2) → N
  44. Intersect(p_1, e_1, 1) → O
  45. Vector(F, N) → v
  46. Vector(F, O) → z
  47. Translate(F, Vector[(z * o)]) → F'_1
  48. Translate(F, Vector[(u * (o - 1))]) → F'_2
  49. Translate(F, Vector[(v * (o - 2))]) → F'_3
  50. Translate(F, Vector[(w_3 * (o - 3))]) → F'
  51. Segment(F, N) → s
  52. LineBisector(s) → a_2
  53. Intersect(a_2, c') → P
  54. CircleArc(P, N, F) → c_2
  55. Ray(P, F'_3) → d_2
  56. Intersect(c_2, d_2, 1) → Q
  57. Segment(F, O) → e_2
  58. LineBisector(e_2) → f_2
  59. Intersect(f_2, a_1) → R
  60. CircleArc(R, F, O) → g_2
  61. Ray(R, F'_1) → h_2
  62. Intersect(g_2, h_2, 1) → S
  63. CircleArc(I, F, J) → k_2
  64. CircleArc(G, H, F) → p_2
  65. Ray(I, F'_2) → i_2
  66. Intersect(k_2, i_2, 1) → T
  67. Ray(G, F') → j_2
  68. Intersect(p_2, j_2, 1) → U
  69. Circle(I, E) → s_1
  70. OrthogonalLine(I, a_1) → i_1
  71. Intersect(s_1, i_1, 2) → V
  72. Intersect(s_1, i_1, 1) → W
  73. OrthogonalLine(V, i_1) → j_1
  74. OrthogonalLine(W, i_1) → k_1
  75. OrthogonalLine(G, k_1) → l_1
  76. Intersect(j_1, l_1) → Z
  77. Intersect(k_1, l_1) → A_1
  78. Midpoint(V, Z) → B_1
  79. Midpoint(A_1, W) → C_1
  80. Circle(B_1, r) → t_1
  81. Circle(C_1, r) → q_2
  82. OrthogonalLine(B_1, k_1) → m_1
  83. Intersect(t_1, m_1, 1) → D_1
  84. Intersect(q_2, m_1, 2) → E_1
  85. Vector(C', Z) → u_1
  86. Vector(B, A_1) → v_1
  87. Vector(N, D_1) → w_1
  88. Vector(O, E_1) → z_1
  89. Translate(N, Vector[(w_1 * (o - 4))]) → N'
  90. Translate(O, Vector[(z_1 * (o - 4))]) → O'
  91. Translate(B, Vector[(v_1 * (o - 4))]) → B'
  92. Translate(A, Vector[(v_1 * (o - 4))]) → A'
  93. Translate(C', Vector[(u_1 * (o - 4))]) → C''
  94. Translate(D, Vector[(u_1 * (o - 4))]) → D'
  95. If(o < 1, S, If[o < 4, O, If[o < 5, O', E_1]]) → F_1
  96. If(o < 1, F, If[o < 2, T, J]) → G_1
  97. If(o < 3, F, If[o < 4, U, H]) → H_1
  98. If(o < 2, F, If[o < 3, Q, If[o < 4, N, If[o < 5, N', D_1]]]) → I_1
  99. If(o < 4, A, If[o < 5, A', W]) → J_1
  100. If(o < 4, B, If[o < 5, B', A_1]) → K_1
  101. If(o < 4, C', If[o < 5, C'', Z]) → L_1
  102. If(o < 4, D, If[o < 5, D', V]) → M_1
  103. Polygon(F_1, J_1, K_1) → poly2, k_3, f_3, j_3
  104. Polygon(L_1, H_1, K_1) → poly3, k_4, l_2, h_3
  105. Polygon(L_1, I_1, M_1) → poly4, m_2, l_3, i_3
  106. Polygon(M_1, G_1, J_1) → poly5, j_4, m_3, g_3
  107. Intersect(m, j_3) → N_1
  108. Segment(I_1, N_1) → n_1
  109. Intersect(j_3, i_3) → O_1
  110. Intersect(c', i_3) → Q_1
  111. Intersect(c', i_3) → R_1
  112. Intersect(c', i_3) → S_1
  113. Intersect(c', i_3) → U_1
  114. Intersect(c', i_3) → A_2
  115. Intersect(l_2, i_3) → B_2
  116. Segment(B_2, O_1) → n_2
  117. Segment(B_2, M_1) → r_2
  118. Midpoint(M_1, L_1) → C_2
  119. Midpoint(L_1, K_1) → D_2
  120. Midpoint(K_1, J_1) → E_2
  121. Midpoint(J_1, M_1) → F_2
  122. Segment(G_1, F_2) → s_2
  123. Segment(C_2, I_1) → t_2
  124. Segment(D_2, H_1) → a_3
  125. Segment(E_2, F_1) → b_3
  126. Polygon(J_1, M_1, L_1, K_1) → poly6, j_5, m_4, l_4, k_5
  127. Intersect(j_3, d_1) → P_1
  128. Segment(P_1, M_1) → c_3
  129. Midpoint(p) → V_1
  130. OrthogonalLine(V_1, p) → d_3
  131. Point(d_3) → W_1
  132. Vector(W_1, V_1) → u_2
  133. Midpoint(r) → T_1
  134. Midpoint(a_3) → Z_1
  135. Point(a_2) → G_2
  136. Vector(G_2, T_1) → v_2
  137. Vector(G_2, Z_1) → w_2
  138. Point(Circle[H_2, w]) → I_2
  139. Segment(H_2, I_2) → e_3
  140. Line(H_2, I_2) → b
  141. Circle(I_2, (3 * w / 4.5)) → p_3
  142. Intersect(p_3, b, 2) → J_2
  143. Rotate(J_2, 30°, I_2) → K_2
  144. Segment(I_2, K_2) → n_3
  145. Line(K_2, b) → q_3
  146. Line(H_2, n_3) → r_3
  147. Intersect(q_3, r_3) → L_2
  148. Segment(H_2, K_2) → s_3
  149. Segment(I_2, L_2) → t_3
  150. Intersect(s_3, t_3) → M_2
  151. Circle(M_2, h) → c_4
  152. OrthogonalLine(M_2, b) → a_4
  153. Intersect(c_4, a_4, 2) → N_2
  154. Polygon(H_2, N_2, I_2) → poly7, i_4, h_4, n_4
  155. Polygon(I_2, N_2, K_2) → poly8, k_6, i_5, n_5
  156. Polygon(H_2, L_2, K_2, I_2) → poly9, h_5, l_6, k_7, i_6
  157. Polygon(H_2, N_2, L_2) → poly10, l_7, h_6, n_6
  158. Polygon(L_2, N_2, K_2) → poly11, k_8, l_8, n_7
  159. Midpoint(k_7) → O_2
  160. Segment(M_2, O_2) → b_4
  161. Segment(M_2, N_2) → d_4
  162. Segment(N_2, O_2) → e_4
  163. Angle(O_2, M_2, N_2) → β
  164. Midpoint(N_2, M_2) → P_2
  165. Midpoint(O_2, N_2) → Q_2
  166. Line(P_2, Q_2) → f_4
  167. Circle(P_2, w) → g_4
  168. Intersect(g_4, f_4, 1) → R_2
  169. Intersect(g_4, f_4, 2) → S_2
  170. Vector(R_2, P_2) → z_2
  171. Vector(S_2, Q_2) → u_3
  172. Midpoint(b_3) → V_2

AI Drawing Prompt Reference

Draw the net of a square pyramid, consisting of a square base and four triangular faces. Start by drawing a square, then draw a circle centered at one vertex of the square. Mark a point on the circle, connect this point to an adjacent vertex of the square, and extend the line to intersect the circle at another point. Draw a smaller circle centered at this new point, intersecting the previous line at two points. Connect these points to the opposite vertex of the square to form four triangles. Finally, connect all the points to complete the net.
The prompt above was compiled from the original drawing commands and is for reference only.