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Polygon Angle Discovery

Polygon Angle Discovery
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2018-11-11 22:43
PolygonAngleGeometric Construction
This example demonstrates the relationship between the number of sides of a polygon and the sum of its interior angles. Key math concepts include polygons, interior angles, angle sum, and mathematical induction. It is constructed using GeoGebra commands like Intersect, Circle, Sequence, Row, and Angle to dynamically create polygons and calculate their interior angles. Suitable for middle school and above students to explore geometric properties and patterns.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Circle(A, B) → c
  3. Sequence(Rotate[B, ((-2 * 3.141592653589793 / Sides) * j), A], j, 1, Sides) → list1
  4. Sequence(Segment[Element[list1, Mod[j, Sides] + 1], Element[list1, Mod[j + 1, Sides] + 1]], j, 1, Sides) → list2
  5. Row(A2) → B2
  6. Sequence(Angle[Element[list1, Mod[j, Sides] + 1], Element[list1, Mod[j + 1, Sides] + 1], Element[list1, Mod[j + 2, Sides] + 1]], j, 1, Sides) → list3
  7. Row(A1) → B1
  8. Angle(Element[list1, Mod[1, Sides] + 1], Element[list1, Mod[2, Sides] + 1], Element[list1, Mod[3, Sides] + 1]) → α
  9. Row(A3) → B3
  10. Row(A4) → B4
  11. Row(A5) → B5
  12. Row(A6) → B6
  13. Row(A7) → B7
  14. Row(A8) → B8
  15. Row(A9) → B9
  16. Row(A10) → B10
  17. Row(A11) → B11
  18. Row(A12) → B12
  19. Row(A13) → B13
  20. Row(A14) → B14
  21. Row(A15) → B15
  22. Row(A16) → B16
  23. Row(A17) → B17
  24. Row(A18) → B18
  25. Row(A19) → B19
  26. Row(A20) → B20
  27. Row(A21) → B21
  28. Row(A22) → B22
  29. Row(A23) → B23
  30. Row(A24) → B24
  31. Row(A25) → B25
  32. Row(A26) → B26
  33. Row(A27) → B27
  34. Row(A28) → B28
  35. Row(A29) → B29
  36. Row(A30) → B30
  37. Row(A31) → B31
  38. Row(A32) → B32
  39. Row(A33) → B33
  40. Row(A34) → B34
  41. Row(A35) → B35
  42. Row(A36) → B36
  43. Angle(Element[list1, Mod[B1, Sides] + 1], Element[list1, Mod[B1 + 1, Sides] + 1], Element[list1, Mod[B1 + 2, Sides] + 1]) → C1
  44. Angle(Element[list1, Mod[B2, Sides] + 1], Element[list1, Mod[B2 + 1, Sides] + 1], Element[list1, Mod[B2 + 2, Sides] + 1]) → C2
  45. Angle(Element[list1, Mod[B3, Sides] + 1], Element[list1, Mod[B3 + 1, Sides] + 1], Element[list1, Mod[B3 + 2, Sides] + 1]) → C3
  46. Angle(Element[list1, Mod[B4, Sides] + 1], Element[list1, Mod[B4 + 1, Sides] + 1], Element[list1, Mod[B4 + 2, Sides] + 1]) → C4
  47. Angle(Element[list1, Mod[B5, Sides] + 1], Element[list1, Mod[B5 + 1, Sides] + 1], Element[list1, Mod[B5 + 2, Sides] + 1]) → C5
  48. Angle(Element[list1, Mod[B6, Sides] + 1], Element[list1, Mod[B6 + 1, Sides] + 1], Element[list1, Mod[B6 + 2, Sides] + 1]) → C6
  49. Angle(Element[list1, Mod[B7, Sides] + 1], Element[list1, Mod[B7 + 1, Sides] + 1], Element[list1, Mod[B7 + 2, Sides] + 1]) → C7
  50. Angle(Element[list1, Mod[B8, Sides] + 1], Element[list1, Mod[B8 + 1, Sides] + 1], Element[list1, Mod[B8 + 2, Sides] + 1]) → C8
  51. Angle(Element[list1, Mod[B9, Sides] + 1], Element[list1, Mod[B9 + 1, Sides] + 1], Element[list1, Mod[B9 + 2, Sides] + 1]) → C9
  52. Angle(Element[list1, Mod[B10, Sides] + 1], Element[list1, Mod[B10 + 1, Sides] + 1], Element[list1, Mod[B10 + 2, Sides] + 1]) → C10
  53. Angle(Element[list1, Mod[B11, Sides] + 1], Element[list1, Mod[B11 + 1, Sides] + 1], Element[list1, Mod[B11 + 2, Sides] + 1]) → C11
  54. Angle(Element[list1, Mod[B12, Sides] + 1], Element[list1, Mod[B12 + 1, Sides] + 1], Element[list1, Mod[B12 + 2, Sides] + 1]) → C12
  55. Angle(Element[list1, Mod[B13, Sides] + 1], Element[list1, Mod[B13 + 1, Sides] + 1], Element[list1, Mod[B13 + 2, Sides] + 1]) → C13
  56. Angle(Element[list1, Mod[B14, Sides] + 1], Element[list1, Mod[B14 + 1, Sides] + 1], Element[list1, Mod[B14 + 2, Sides] + 1]) → C14
  57. Angle(Element[list1, Mod[B15, Sides] + 1], Element[list1, Mod[B15 + 1, Sides] + 1], Element[list1, Mod[B15 + 2, Sides] + 1]) → C15
  58. Angle(Element[list1, Mod[B16, Sides] + 1], Element[list1, Mod[B16 + 1, Sides] + 1], Element[list1, Mod[B16 + 2, Sides] + 1]) → C16
  59. Angle(Element[list1, Mod[B17, Sides] + 1], Element[list1, Mod[B17 + 1, Sides] + 1], Element[list1, Mod[B17 + 2, Sides] + 1]) → C17
  60. Angle(Element[list1, Mod[B18, Sides] + 1], Element[list1, Mod[B18 + 1, Sides] + 1], Element[list1, Mod[B18 + 2, Sides] + 1]) → C18
  61. Angle(Element[list1, Mod[B19, Sides] + 1], Element[list1, Mod[B19 + 1, Sides] + 1], Element[list1, Mod[B19 + 2, Sides] + 1]) → C19
  62. Angle(Element[list1, Mod[B20, Sides] + 1], Element[list1, Mod[B20 + 1, Sides] + 1], Element[list1, Mod[B20 + 2, Sides] + 1]) → C20
  63. Angle(Element[list1, Mod[B21, Sides] + 1], Element[list1, Mod[B21 + 1, Sides] + 1], Element[list1, Mod[B21 + 2, Sides] + 1]) → C21
  64. Angle(Element[list1, Mod[B22, Sides] + 1], Element[list1, Mod[B22 + 1, Sides] + 1], Element[list1, Mod[B22 + 2, Sides] + 1]) → C22
  65. Angle(Element[list1, Mod[B23, Sides] + 1], Element[list1, Mod[B23 + 1, Sides] + 1], Element[list1, Mod[B23 + 2, Sides] + 1]) → C23
  66. Angle(Element[list1, Mod[B24, Sides] + 1], Element[list1, Mod[B24 + 1, Sides] + 1], Element[list1, Mod[B24 + 2, Sides] + 1]) → C24
  67. Angle(Element[list1, Mod[B25, Sides] + 1], Element[list1, Mod[B25 + 1, Sides] + 1], Element[list1, Mod[B25 + 2, Sides] + 1]) → C25
  68. Angle(Element[list1, Mod[B26, Sides] + 1], Element[list1, Mod[B26 + 1, Sides] + 1], Element[list1, Mod[B26 + 2, Sides] + 1]) → C26
  69. Angle(Element[list1, Mod[B27, Sides] + 1], Element[list1, Mod[B27 + 1, Sides] + 1], Element[list1, Mod[B27 + 2, Sides] + 1]) → C27
  70. Angle(Element[list1, Mod[B28, Sides] + 1], Element[list1, Mod[B28 + 1, Sides] + 1], Element[list1, Mod[B28 + 2, Sides] + 1]) → C28
  71. Angle(Element[list1, Mod[B29, Sides] + 1], Element[list1, Mod[B29 + 1, Sides] + 1], Element[list1, Mod[B29 + 2, Sides] + 1]) → C29
  72. Angle(Element[list1, Mod[B30, Sides] + 1], Element[list1, Mod[B30 + 1, Sides] + 1], Element[list1, Mod[B30 + 2, Sides] + 1]) → C30
  73. Angle(Element[list1, Mod[B31, Sides] + 1], Element[list1, Mod[B31 + 1, Sides] + 1], Element[list1, Mod[B31 + 2, Sides] + 1]) → C31
  74. Angle(Element[list1, Mod[B32, Sides] + 1], Element[list1, Mod[B32 + 1, Sides] + 1], Element[list1, Mod[B32 + 2, Sides] + 1]) → C32
  75. Angle(Element[list1, Mod[B33, Sides] + 1], Element[list1, Mod[B33 + 1, Sides] + 1], Element[list1, Mod[B33 + 2, Sides] + 1]) → C33
  76. Angle(Element[list1, Mod[B34, Sides] + 1], Element[list1, Mod[B34 + 1, Sides] + 1], Element[list1, Mod[B34 + 2, Sides] + 1]) → C34
  77. Angle(Element[list1, Mod[B35, Sides] + 1], Element[list1, Mod[B35 + 1, Sides] + 1], Element[list1, Mod[B35 + 2, Sides] + 1]) → C35
  78. Angle(Element[list1, Mod[B36, Sides] + 1], Element[list1, Mod[B36 + 1, Sides] + 1], Element[list1, Mod[B36 + 2, Sides] + 1]) → C36
  79. Sequence(Polygon[Element[list1, Sides], Element[list1, j], Element[list1, j + 1]], j, 1, Sides - 2) → list4