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Net of a Cone and Its Surface Area Formula

Net of a Cone and Its Surface Area Formula
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2018-03-25 23:58
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This GeoGebra applet dynamically demonstrates the net of a cone, showing how the cone's surface unfolds into a circular sector and a base circle, and how the total surface area is composed of the lateral area and the base area. Core mathematical concepts include the geometry of a cone, sector area formula, relationship between arc length and circumference, and derivation of the surface area formula. Construction highlights: using GeoGebra commands such as Point, Circle, Intersect, Ellipse, Arc, Segment, Vector, Translate, Rotate, Mirror, Angle, Line, Midpoint, If, Conic, OrthogonalLine, Tangent, and CircleSector to build the cone, create the unfolding animation, and annotate measurements. Suitable for middle and high school geometry classes to explore 3D-to-2D nets and surface area calculations.

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'_1
  10. Translate(e, Vector[u * h]) → e'_1
  11. Arc(e'_1, C'_1, B'_1) → p_5
  12. Arc(e'_1, B'_1, C'_1) → q
  13. Segment(A'_1, A) → h_6
  14. Segment(A'_1, C'_1) → p
  15. Angle(A, A'_1, C'_1) → α
  16. Line(A, A'_1) → j
  17. Intersect(e'_1, j) → H, I
  18. Circle(I, r) → t
  19. Intersect(t, j, 2) → K
  20. Circle(K, I) → d_1
  21. Intersect(d_1, j, 2) → N
  22. Vector(H, N) → v
  23. Translate(H, Vector[v * n]) → H'_1
  24. Translate(D, Vector[u * h]) → D'
  25. Translate(E, Vector[u * h]) → E'
  26. Line(B, B'_1) → l
  27. Line(C, C'_1) → m
  28. Intersect(d_1, l, 1) → P
  29. Intersect(d_1, m, 1) → Q
  30. Vector(B'_1, P) → z
  31. Vector(C'_1, Q) → u_1
  32. Translate(B'_1, Vector[z * n]) → B''
  33. Translate(C'_1, Vector[u_1 * n]) → C''
  34. Midpoint(B'_1, A'_1) → R
  35. Midpoint(B, A) → S
  36. Line(R, S) → a_1
  37. Intersect(q, a_1) → T,
  38. Intersect(d_1, a_1, 1) → U
  39. Vector(T, U) → v_1
  40. Translate(T, Vector[v_1 * n]) → T'
  41. If(n ≟ 0.01, B'_1, If[n < 1, B'', P]) → B_1
  42. If(n ≟ 0.01, T, If[n < 1, T', U]) → D_1
  43. If(n ≟ 0.01, H, If[n < 1, H'_1, N]) → E_1
  44. If(n ≟ 0.01, C'_1, If[n < 1, C'', Q]) → F_1
  45. Conic(B_1, D_1, E_1, F_1, I) → g_1
  46. Conic(I, B_1, D_1, E_1, F_1) → h_1
  47. OrthogonalLine(I, j) → l_1
  48. OrthogonalLine(H, j) → b_2
  49. Intersect(m, l_1) → H_2
  50. Midpoint(H_2, C'_1) → I_2
  51. OrthogonalLine(C'_1, m) → i_2
  52. Segment(K, F_1) → m_4
  53. Translate(e, Vector[u * h]) → e'
  54. Tangent(A, e'_1) → a, b
  55. Intersect(e', b) → J, G_2
  56. Intersect(e', a) → L, R_2
  57. Segment(A, J) → f
  58. Segment(A, L) → g
  59. Circle(I, sqrt(h^(2) + r^(2))) → k
  60. Intersect(k, j, 1) → M
  61. Circle(M, sqrt(h^(2) + r^(2))) → s
  62. Segment(M, I) → i
  63. Rotate(I, γ, M) → I'_1
  64. Rotate(I, -γ, M) → I'_2
  65. OrthogonalLine(I'_2, i_2) → b_1
  66. Intersect(b_2, b_1) → O
  67. Midpoint(H, O) → V
  68. Circle(V, O) → c_1
  69. Vector(O, I'_2) → w
  70. Center(g_1) → W
  71. Segment(W, F_1) → e_1
  72. Vector(H, A) → w_1
  73. Translate(H, Vector[w_1 * (n - 1)]) → H'
  74. If(n < 1, H, If[n < 2, H', A]) → Z
  75. Segment(H, Z) → f_1
  76. Rotate(I, -δ, M) → I'
  77. Vector(L, I') → z_1
  78. Translate(c_1, Vector[w * (n - 2)]) → k_1
  79. Center(k_1) → A_1
  80. Intersect(k_1, j) → C_1, Q_1
  81. Rotate(C_1, 180° * (n - 2), A_1) → G_1
  82. If(n < 2, H, If[n ≤ 3, G_1, I'_2]) → H_1
  83. Mirror(H_1, j) → I_1
  84. Translate(L, Vector[z_1 * (n - 2)]) → L'
  85. If(n < 2, L, If[n ≤ 3, L', I']) → J_1
  86. Mirror(J_1, j) → K_1
  87. Conic(I_1, H_1, J_1, I, K_1) → p_1
  88. Vector(A, M) → i_1
  89. Translate(A, Vector[i_1 * (n - 2)]) → A'
  90. If(n < 2, A, If[n ≤ 3, A', M]) → L_1
  91. Tangent(L_1, p_1) → j_1, m_1
  92. Intersect(p_1, m_1, 1) → M_1
  93. Intersect(p_1, j_1, 1) → N_1
  94. Polygon(L_1, M_1, N_1) → poly1, n_1, l_2, m_2
  95. Arc(p_1, N_1, M_1) → q_1
  96. Line(L_1, b_2) → r_1
  97. Intersect(p_1, r_1, 1) → O_1
  98. Intersect(p_1, r_1, 2) → P_1
  99. Arc(p_1, P_1, O_1) → s_1
  100. Polygon(L_1, O_1, I_1) → poly2, i_3, l_3, o_1
  101. Polygon(L_1, H_1, P_1) → poly3, p_2, l_4, h_2
  102. Arc(p_1, O_1, I_1) → t_1
  103. Arc(p_1, H_1, P_1) → c_2
  104. Segment(L_1, I_1) → a_2
  105. Segment(L_1, H_1) → d_2
  106. Arc(p_1, H_1, I_1) → e_2
  107. Polygon(I_1, L_1, H_1) → poly4, h_3, i_4, l_5
  108. CircleSector(M, I'_2, I'_1) → f_2
  109. CircleSector(M, I'_2, I'_1) → g_2
  110. Arc(p_1, H_1, I_1) → k_2
  111. Segment(M_1, L_1) → j_2
  112. Segment(N_1, L_1) → n_2
  113. Polygon(I_1, L_1, H_1) → poly5, h_4, i_5, l_6
  114. Arc(p_1, H_1, I_1) → q_2
  115. Polygon(J, A, L) → poly6, l_7, j_3, a_3
  116. Point(xAxis) → R_1
  117. OrthogonalLine(R_1, xAxis) → r_2
  118. Intersect(i_2, r_2) → S_1
  119. Circle(R_1, r) → s_2
  120. Intersect(s_2, xAxis) → T_1, U_1
  121. OrthogonalLine(T_1, i_2) → t_2
  122. OrthogonalLine(V_1, i_2) → b_3
  123. OrthogonalLine(W_1, i_2) → c_3
  124. Intersect(i_2, t_2) → Z_1
  125. Intersect(i_2, b_3) → A_2
  126. Intersect(i_2, c_3) → B_2
  127. Ellipse(A_2, B_2, Z_1) → d_3
  128. Tangent(R_1, d_3) → e_3, f_3
  129. Intersect(d_3, f_3, 1) → C_2
  130. Intersect(d_3, e_3, 1) → D_2
  131. Polygon(C_2, D_2, R_1) → poly7, r_3, c_4, d_4
  132. Ellipse(A_2, B_2, Z_1) → g_3
  133. OrthogonalLine(U_1, xAxis) → k_3
  134. Intersect(g_3, k_3) → E_2, J_2
  135. Segment(S_1, E_2) → m_3
  136. Segment(S_1, R_1) → n_3
  137. Intersect(d_3, m_3, 1) → F_2
  138. Midpoint(g) → K_2
  139. Midpoint(f_1) → L_2
  140. Point(j) → M_2
  141. Midpoint(I_1, A') → N_2
  142. Midpoint(A', H_1) → O_2
  143. Vector(M_2, N_2) → u_2
  144. Vector(M_2, O_2) → v_2
  145. Line(M_2, r_1) → p_3
  146. Point(p_3) → P_2
  147. Vector(P_2, K_2) → z_2
  148. Vector(P_2, L_2) → u_3
  149. Center(g_3) → Q_2
  150. Angle(F_2, Q_2, R_1) → ε
  151. Segment(I_1, H_1) → q_3
  152. Intersect(k_2, q_3, 2) → S_2
  153. Point(poly7) → T_2