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Interior and Exterior Angles in Regular Polygons by Nicola Trubridge

Interior and Exterior Angles in Regular Polygons by Nicola Trubridge
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2015-10-01 03:18
PolygonAngleGeometric Construction
In collections GCSE Higher Ch5 Angles & Trigonometry
This example demonstrates the calculation and relationship between interior and exterior angles of regular polygons. Key math concepts include the sum of interior angles formula, exterior angle theorem, and trigonometric function applications. GeoGebra commands used are Circle, Segment, Point, Rotate, and Angle for construction. Suitable for high school mathematics teaching to understand geometric properties and calculations of regular polygons.

GeoGebra Commands & Parameters

  1. Circle(M_7, N_7) → r_7
  2. Segment(M_7, N_7) → s_6
  3. Segment(M_7, N_7) → a_6
  4. Point(a_6) → C_3
  5. Rotate(N_7, 240*°, C_3) → E_1
  6. Angle(E_1, C_3, N_7) → α_1
  7. Rotate(E_1, 240*°, C_3) → E'_1
  8. Segment(E_1, C_3) → f_4
  9. Segment(C_3, E'_1) → g_4
  10. Segment(E_1, N_7) → h_4
  11. Segment(E'_1, E_1) → i_4
  12. Segment(E'_1, N_7) → j_4
  13. Angle(N_7, E_1, E'_1) → θ_2
  14. Angle(E'_1, E_1, K_1) → ο_1
  15. Segment(E_1, K_1) → n_4
  16. Circle(M_{10}, N_8) → r_8
  17. Segment(M_{10}, N_8) → s_2
  18. Rotate(N_8, 300*°, M_{10}) → N'_3
  19. Segment(M_{10}, N'_3) → a_7
  20. Rotate(N'_3, 300*°, M_{10}) → O_2
  21. Angle(O_2, M_{10}, N'_3) → γ_2
  22. Segment(M_{10}, O_2) → b_5
  23. Segment(O_2, N'_3) → g_5
  24. Segment(N'_3, N_8) → h_5
  25. Angle(N_8, N'_3, O_2) → ε_1
  26. Circle(M_{15}, N_3) → r_3
  27. Segment(M_{15}, N_3) → s_3
  28. Segment(M_{15}, N_3) → a_1
  29. Rotate(N_3, 288*°, M_{15}) → A_2
  30. Segment(M_{15}, A_2) → b_1
  31. Rotate(A_2, 288*°, M_{15}) → A'_2
  32. Segment(M_{15}, A'_2) → c_1
  33. Rotate(A'_2, 288*°, M_{15}) → A''_2
  34. Angle(A''_2, M_{15}, A'_2) → θ_3
  35. Segment(M_{15}, A''_2) → d_1
  36. Rotate(A''_2, 288*°, M_{15}) → A'''_2
  37. Segment(M_{15}, A'''_2) → e_1
  38. Segment(A'''_2, N_3) → f_1
  39. Segment(N_3, A_2) → g_1
  40. Segment(A_2, A'_2) → h_1
  41. Segment(A'_2, A''_2) → i_1
  42. Segment(A''_2, A'''_2) → j_1
  43. Angle(N_3, A_2, A'_2) → γ_1
  44. Angle(A'_2, A_2, F_1) → κ_1
  45. Segment(A_2, F_1) → k_1
  46. Circle(M_{16}, N_1) → r_1
  47. Segment(M_{16}, N_1) → s_9
  48. Segment(M_{16}, N_1) → a_3
  49. Point(a_3) → C_1
  50. Rotate(N_1, 270*°, C_1) → D_1
  51. Rotate(D_1, 270*°, C_1) → D'_1
  52. Angle(D'_1, C_1, D_1) → β_1
  53. Rotate(D'_1, 270*°, C_1) → D''_1
  54. Segment(D_1, C_1) → p_1
  55. Segment(D'_1, C_1) → q_1
  56. Segment(D''_1, C_1) → r_9
  57. Segment(D'_1, D_1) → s_1
  58. Segment(D_1, N_1) → t_2
  59. Segment(N_1, D''_1) → d_2
  60. Segment(D''_1, D'_1) → e_2
  61. Angle(N_1, D_1, D'_1) → ι_1
  62. Angle(D'_1, D_1, I_1) → ν_1
  63. Segment(D_1, I_1) → q_2
  64. Circle(M_{17}, N_2) → r_2
  65. Segment(M_{17}, N_2) → s_5
  66. Rotate(N_2, 60*°, M_{17}) → N'_1
  67. Segment(N'_1, M_{17}) → t_1
  68. Rotate(N_2, 300*°, M_{17}) → N'_2
  69. Segment(M_{17}, N'_2) → a_2
  70. Rotate(N'_2, 300*°, M_{17}) → O_1
  71. Angle(O_1, M_{17}, N'_2) → γ_3
  72. Segment(M_{17}, O_1) → b_2
  73. Rotate(O_1, 300*°, M_{17}) → O'_1
  74. Segment(M_{17}, O'_1) → c_2
  75. Rotate(O'_1, 300*°, M_{17}) → O''_1
  76. Segment(O''_1, M_{17}) → d_3
  77. Segment(O''_1, O'_1) → e_3
  78. Segment(O'_1, O_1) → f_2
  79. Segment(O_1, N'_2) → g_2
  80. Segment(N'_2, N_2) → h_2
  81. Segment(N_2, N'_1) → i_2
  82. Segment(N'_1, O''_1) → j_2
  83. Angle(N_2, N'_2, O_1) → ε_2
  84. Angle(O_1, N'_2, G_1) → λ_1
  85. Segment(N'_2, G_1) → l_1
  86. Circle(M_{18}, N_4) → r_4
  87. Segment(M_{18}, N_4) → s_4
  88. Segment(M_{18}, N_4) → a_4
  89. Rotate(N_4, 308.57*°, M_{18}) → A_1
  90. Rotate(A_1, 308.57*°, M_{18}) → A'_1
  91. Rotate(A'_1, 308.57*°, M_{18}) → A''_1
  92. Rotate(A''_1, 308.57*°, M_{18}) → A'''_1
  93. Angle(A'''_1, M_{18}, A''_1) → δ_1
  94. Rotate(A'''_1, 308.57*°, M_{18}) → B_1
  95. Rotate(B_1, 308.57*°, M_{18}) → B'_1
  96. Segment(M_{18}, A_1) → b_3
  97. Segment(M_{18}, A'_1) → c_3
  98. Segment(M_{18}, A''_1) → d_4
  99. Segment(M_{18}, A'''_1) → e_4
  100. Segment(M_{18}, B_1) → f_3
  101. Segment(M_{18}, B'_1) → g_3
  102. Segment(A_1, A'_1) → h_3
  103. Segment(A'_1, A''_1) → i_3
  104. Segment(A''_1, A'''_1) → j_3
  105. Segment(A'''_1, B_1) → k_2
  106. Segment(B_1, B'_1) → l_2
  107. Segment(B'_1, N_4) → m_1
  108. Segment(N_4, A_1) → n_1
  109. Angle(A_1, A'_1, A''_1) → η_1
  110. Angle(A''_1, A'_1, J_1) → ξ_1
  111. Segment(A'_1, J_1) → p_2
  112. Circle(M_{19}, N_5) → r_5
  113. Segment(M_{19}, N_5) → s_8
  114. Segment(M_{19}, N_5) → k_3
  115. Rotate(N_5, 315*°, M_{19}) → B_3
  116. Angle(B_3, M_{19}, N_5) → κ_3
  117. Segment(M_{19}, B_3) → l_3
  118. Rotate(B_3, 315*°, M_{19}) → B'_3
  119. Segment(B'_3, M_{19}) → m_2
  120. Rotate(B'_3, 315*°, M_{19}) → B''_1
  121. Rotate(B''_1, 315*°, M_{19}) → B'''_1
  122. Rotate(B'''_1, 315*°, M_{19}) → C_4
  123. Rotate(C_4, 315*°, M_{19}) → C'_1
  124. Rotate(C'_1, 315*°, M_{19}) → C''_1
  125. Segment(B''_1, M_{19}) → n_2
  126. Segment(M_{19}, C''_1) → p_3
  127. Segment(C'_1, M_{19}) → q_3
  128. Segment(C_4, M_{19}) → k_6
  129. Segment(B'''_1, M_{19}) → l_6
  130. Segment(B''_1, B'''_1) → m_3
  131. Segment(B'''_1, C_4) → n_6
  132. Segment(C_4, C'_1) → p_6
  133. Segment(C'_1, C''_1) → q_6
  134. Segment(C''_1, N_5) → t_4
  135. Segment(N_5, B_3) → a_8
  136. Segment(B_3, B'_3) → b_6
  137. Segment(B'_3, B''_1) → c_5
  138. Angle(B_3, B'_3, B''_1) → ζ_2
  139. Angle(B''_1, B'_3, H_2) → μ_2
  140. Segment(B'_3, H_2) → m_7

AI Drawing Prompt Reference

Draw a regular polygon with its interior and exterior angles. Start by drawing a circle, then create two line segments from a point on the circle to form the sides of the polygon. Rotate these segments around the point to construct the other sides of the polygon, and mark the interior and exterior angles.
The prompt above was compiled from the original drawing commands and is for reference only.