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Interactive Geometry of Circle Circumference and Radius

Interactive Geometry of Circle Circumference and Radius
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2024-01-25 03:18
CircleAngleVectorTransformationGeometric Construction
This example demonstrates how a circle’s circumference is represented and how the key radius-related measures correspond as the circle’s scale changes. It centers on fundamental circle concepts: radius, circle arcs and perimeter (circumference), and the role of angles (especially central angles) in relating geometric quantities. Construction highlights include creating segments that represent the radius (e.g., r and related objects) with Segment, building the circle and relevant arcs with Circle and CircleArc, and defining and locating critical points using Point and Intersect. Auxiliary lines are formed with Line and OrthogonalLine to support angle or distance construction. The model uses geometric transformations such as Rotate, Translate, Dilate, and Mirror to generate corresponding objects (e.g., primed duplicates) while preserving the intended geometric relationships. Angle markers are added via Angle (e.g., α) to connect the construction to central-angle geometry. Conditional logic (If) and vectors (Vector) are used to drive interactive behavior and linked annotations. Suitable for middle school geometry and early high school instruction, as well as exploratory learning and classroom demonstration.

GeoGebra Commands & Parameters

  1. Segment((0, 0), (r, 0)) → f
  2. Rotate((r, 0), 1, (0, 0)) → A
  3. Rotate((r, 0), 2, (0, 0)) → B
  4. Rotate((r, 0), 3, (0, 0)) → C
  5. Rotate((r, 0), 4, (0, 0)) → D
  6. Rotate((r, 0), 5, (0, 0)) → E
  7. Rotate((r, 0), -1, (0, 0)) → F
  8. Rotate(f, If[0 < a ≤ 1, ((-a) * pi / 2), ((-pi)) / 2], (r, 0)) → f'
  9. Rotate(G, If[1.5 < a ≤ 2.5, a - 1.5, 1], (0, 0)) → G'
  10. CircleArc((0, 0), G, G') → c
  11. Circle(G', r - c) → d
  12. Line((0, 0), G') → g
  13. OrthogonalLine(G', g) → h
  14. Intersect(d, h, 2) → H
  15. Segment(G', H) → i
  16. Dilate(G, If[3 < a ≤ 4, 4 - a, 0], A) → G'_{1}
  17. Vector(G, G'_{1}) → u
  18. Translate(f, u) → f'_{1}
  19. Line((0, 0), A) → j
  20. OrthogonalLine(A, j) → k
  21. Point(p) → I
  22. Point(k) → J
  23. Angle(J, A, I) → α
  24. Rotate(f'_{1}, If[3 < a ≤ 4, ((-(a - 3)) * α), (-α)], G'_{1}) → l
  25. Rotate(A, If[4 < a ≤ 5, a - 4, 1], (0, 0)) → A'
  26. CircleArc((0, 0), A, A') → e
  27. Circle(A', r - e) → q
  28. Line((0, 0), A') → m
  29. OrthogonalLine(A', m) → n
  30. Intersect(q, n, 2) → K
  31. Segment(A', K) → s
  32. Dilate((0, 0), If[5.5 < a ≤ 6.5, 6.5 - a, 0], B) → L
  33. Vector(M, L) → v
  34. Translate(f, v) → f'_{2}
  35. Line(M, B) → b
  36. OrthogonalLine(B, b) → f_1
  37. Point(t) → N
  38. Point(f_1) → O
  39. Angle(O, B, N) → β
  40. Rotate(f'_{2}, If[5.5 < a ≤ 6.5, ((-(a - 5.5)) * β), (-β)], L) → g_1
  41. Rotate(B, If[6.5 < a ≤ 7.5, a - 6.5, 1], (0, 0)) → B'
  42. CircleArc(M, B, B') → c_1
  43. Line(M, B') → h_1
  44. OrthogonalLine(B', h_1) → i_1
  45. Circle(B', r - c_1) → d_1
  46. Intersect(d_1, i_1, 2) → P
  47. Dilate(M, If[8 < a ≤ 9, 9 - a, 0], C) → M'
  48. Vector(M, M') → w
  49. Translate(f, w) → f'_{3}
  50. Line(M, C) → k_1
  51. OrthogonalLine(C, k_1) → l_1
  52. Point(p_1) → Q
  53. Point(l_1) → R
  54. Angle(R, C, Q) → γ
  55. Rotate(f'_{3}, If[8 < a ≤ 9, ((-(a - 8)) * γ), (-γ)], M') → m_1
  56. Rotate(C, If[9 < a ≤ 10, a - 9, 1], M) → C'
  57. CircleArc(M, C, C') → e_1
  58. Line(M, C') → n_1
  59. OrthogonalLine(C', n_1) → q_1
  60. Circle(C', r - e_1) → r_1
  61. Intersect(r_1, q_1) → S, B_1
  62. Intersect(d_1, i_1, 1) → T
  63. Dilate(M, If[10.5 < a ≤ 11.5, 11.5 - a, 0], D) → M'_{1}
  64. Vector(M, M'_{1}) → u_1
  65. Translate(f, u_1) → f'_{4}
  66. Line(M, D) → a_1
  67. OrthogonalLine(D, a_1) → b_1
  68. Point(b_1) → U
  69. Point(t_1) → V
  70. Angle(U, D, V) → δ
  71. Rotate(f'_{4}, If[10.5 < a ≤ 11.5, ((-(a - 10.5)) * δ), (-δ)], M'_{1}) → f_2
  72. Rotate(D, If[11.5 < a ≤ 12.5, a - 11.5, 1], M) → D'
  73. CircleArc(M, D, D') → c_2
  74. Line(M, D') → g_2
  75. OrthogonalLine(D', g_2) → h_2
  76. Circle(D', r - c_2) → d_2
  77. Intersect(d_2, h_2, 2) → W
  78. Segment(D', W) → i_2
  79. Intersect(d_2, h_2, 1) → Z
  80. Rotate(M, If[13 < a ≤ 14, ((a - 13) * pi / 2), pi / 2], G) → M'_{2}
  81. Segment(G, M'_{2}) → j_2
  82. If(x(P) ≤ x(T), P, T) → A_1
  83. Segment(B', A_1) → j_1
  84. If(y(B_1) < y(S), B_1, S) → C_1
  85. Segment(C', C_1) → s_1
  86. Rotate(G, If[14 < a ≤ 15, (-(a - 14)), -1], M) → G'_{2}
  87. CircleArc(M, G'_{2}, G) → e_2
  88. Line(M, G'_{2}) → k_2
  89. Circle(G'_{2}, r - e_2) → p_2
  90. OrthogonalLine(G'_{2}, k_2) → l_2
  91. Intersect(p_2, l_2) → D_1, E_1
  92. If(y(D_1) ≤ y(E_1), D_1, E_1) → F_1
  93. Segment(G'_{2}, F_1) → m_2
  94. Rotate(D, 1, M) → D'_{1}
  95. Rotate(G, 0.5) → G'_{3}
  96. Rotate(G, 1.5) → G'_{4}
  97. Rotate(G, 2.5) → G'_{5}
  98. Rotate(G, 3.5) → G'_{6}
  99. Rotate(G, 4.5) → G'_{7}
  100. Rotate(G, -0.5) → G'_{8}
  101. Segment(G_1, G) → n_2
  102. Circle(G_1, n_2 / 7) → q_2
  103. Intersect(q_2, n_2, 1) → H_1
  104. Mirror(G_1, H_1) → I_1
  105. Mirror(H_1, I_1) → J_1
  106. Mirror(I_1, J_1) → K_1
  107. Mirror(J_1, K_1) → L_1
  108. Mirror(K_1, L_1) → M_1
  109. Dilate(G_1, If[16 < a ≤ 17, 17 - a, 0], H_1) → N_1
  110. Segment(G_1, N_1) → r_2
  111. Rotate(G_1, If[17 < a ≤ 18, ((a - 17) * pi), pi], H_1) → O_1
  112. Rotate(N_1, If[17 < a ≤ 18, ((a - 17) * pi), pi], H_1) → P_1
  113. Segment(O_1, P_1) → s_2
  114. Rotate(O_1, If[18 < a ≤ 19, ((a - 18) * pi), pi], I_1) → Q_1
  115. Rotate(P_1, If[18 < a ≤ 19, ((a - 18) * pi), pi], I_1) → R_1
  116. Segment(Q_1, R_1) → t_2
  117. Rotate(Q_1, If[19 < a ≤ 20, ((a - 19) * pi), pi], J_1) → S_1
  118. Rotate(R_1, If[19 < a ≤ 20, ((a - 19) * pi), pi], J_1) → T_1
  119. Segment(S_1, T_1) → a_2
  120. Rotate(S_1, If[20 < a ≤ 21, ((a - 20) * pi), pi], K_1) → U_1
  121. Rotate(T_1, If[20 < a ≤ 21, ((a - 20) * pi), pi], K_1) → V_1
  122. Segment(U_1, V_1) → b_2
  123. Rotate(U_1, If[21 < a ≤ 22, ((a - 21) * pi), pi], L_1) → W_1
  124. Rotate(V_1, If[21 < a ≤ 22, ((a - 21) * pi), pi], L_1) → Z_1
  125. Segment(W_1, Z_1) → f_3
  126. Rotate(W_1, If[22 < a ≤ 23, ((a - 22) * pi), pi], M_1) → A_2
  127. Rotate(Z_1, If[22 < a ≤ 23, ((a - 22) * pi), pi], M_1) → B_2
  128. Segment(A_2, B_2) → g_3
  129. Dilate(I_1, If[24 < a ≤ 25, 25 - a, 0], F) → C_2
  130. Vector(I_1, C_2) → v_1
  131. Translate(O_1, v_1) → D_2
  132. Translate(P_1, v_1) → E_2
  133. Segment(D_2, E_2) → h_3
  134. Translate(G_1, v_1) → F_2
  135. Translate(N_1, v_1) → G_2
  136. Segment(F_2, G_2) → i_3
  137. Line(G_1, F) → j_3
  138. OrthogonalLine(F, j_3) → k_3
  139. Line(F, n_2) → l_3
  140. Point(l_3) → H_2
  141. Point(k_3) → I_2
  142. Angle(H_2, F, I_2) → ε
  143. Rotate(D_2, If[24 < a ≤ 25, ((a - 24) * ε), ε], D_2) → J_2
  144. Rotate(E_2, If[24 < a ≤ 25, ((a - 24) * ε), ε], D_2) → K_2
  145. Rotate(F_2, If[24 < a ≤ 25, ((a - 24) * ε), ε], D_2) → L_2
  146. Rotate(G_2, If[24 < a ≤ 25, ((a - 24) * ε), ε], D_2) → M_2
  147. Segment(J_2, L_2) → m_3
  148. Rotate(F, If[25 < a ≤ 26, ((-(a - 25)) * ((2 * pi) - 6)), 6 - (2 * pi)], G_1) → F'
  149. CircleArc(G_1, F', F) → c_3
  150. Line(G_1, F') → n_3
  151. OrthogonalLine(F', n_3) → p_3
  152. Intersect(p_3, eq3) → N_2
  153. Ray(F', N_2) → q_3
  154. Point(q_3) → O_2
  155. CircleArc(G_1, D'_{1}, F) → d_3
  156. Circle(F', d_3 - c_3) → e_3
  157. Intersect(e_3, q_3, 1) → P_2
  158. Segment(F', P_2) → r_3

AI Drawing Prompt Reference

Draw a circle to illustrate the relationship between its radius, arcs, and circumference. Start with a horizontal segment from the center as the radius, then rotate it multiple times around the center to get equally spaced points. Connect these points to form the circle's circumference and draw arcs between them. Use rotations and scaling to create multiple copies of the circle, and construct angle relationships with auxiliary and perpendicular lines. Label key angles and use vectors to drive translations of certain objects. The final image should clearly show the geometric connections between radius, arcs, circumference, and angles, suitable for educational purposes.
The prompt above was compiled from the original drawing commands and is for reference only.