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Relationship Between Apothem and Perimeter of Regular Polygons

Relationship Between Apothem and Perimeter of Regular Polygons
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2013-07-02 21:53
PolygonVectorGeometric Construction
In collections Regular Polygons
This example explores the relationship between the apothem and perimeter of regular polygons with 3 to 40 sides. By adjusting the number of sides, observe how the apothem affects the perimeter. Key mathematical concepts include regular polygons, apothem, and perimeter. GeoGebra commands used include Rotate, If, Sequence, Distance, Segment, Polygon, and Vector. Suitable for students in middle school and above to learn about polygon properties.

GeoGebra Commands & Parameters

  1. Rotate(Pt1, -α, Pt2) → Pt3R
  2. Rotate(Pt2, -α, Pt3R) → Pt4R
  3. Rotate(Pt3R, -α, Pt4R) → Pt5R
  4. Rotate(Pt4R, -α, Pt5R) → Pt6R
  5. Rotate(Pt5R, -α, Pt6R) → Pt7R
  6. Rotate(Pt6R, -α, Pt7R) → Pt8R
  7. Rotate(Pt7R, -α, Pt8R) → Pt9R
  8. Rotate(Pt8R, -α, Pt9R) → Pt10R
  9. Rotate(Pt9R, -α, Pt10R) → Pt11R
  10. Rotate(Pt10R, -α, Pt11R) → Pt12R
  11. Rotate(Pt11R, -α, Pt12R) → Pt13R
  12. Rotate(Pt12R, -α, Pt13R) → Pt14R
  13. Rotate(Pt13R, -α, Pt14R) → Pt15R
  14. Rotate(Pt14R, -α, Pt15R) → Pt16R
  15. Rotate(Pt15R, -α, Pt16R) → Pt17R
  16. Rotate(Pt16R, -α, Pt17R) → Pt18R
  17. Rotate(Pt17R, -α, Pt18R) → Pt19R
  18. Rotate(Pt18R, -α, Pt19R) → Pt20R
  19. Rotate(Pt19R, -α, Pt20R) → Pt21R
  20. Rotate(Pt20R, -α, Pt21R) → Pt22R
  21. Rotate(Pt2, α, Pt1) → Pt3L
  22. Rotate(Pt1, α, Pt3L) → Pt4L
  23. Rotate(Pt3L, α, Pt4L) → Pt5L
  24. Rotate(Pt4L, α, Pt5L) → Pt6L
  25. Rotate(Pt5L, α, Pt6L) → Pt7L
  26. Rotate(Pt6L, α, Pt7L) → Pt8L
  27. Rotate(Pt7L, α, Pt8L) → Pt9L
  28. Rotate(Pt8L, α, Pt9L) → Pt10L
  29. Rotate(Pt9L, α, Pt10L) → Pt11L
  30. Rotate(Pt10L, α, Pt11L) → Pt12L
  31. Rotate(Pt11L, α, Pt12L) → Pt13L
  32. Rotate(Pt12L, α, Pt13L) → Pt14L
  33. Rotate(Pt13L, α, Pt14L) → Pt15L
  34. Rotate(Pt14L, α, Pt15L) → Pt16L
  35. Rotate(Pt15L, α, Pt16L) → Pt17L
  36. Rotate(Pt16L, α, Pt17L) → Pt18L
  37. Rotate(Pt17L, α, Pt18L) → Pt19L
  38. Rotate(Pt18L, α, Pt19L) → Pt20L
  39. Rotate(Pt19L, α, Pt20L) → Pt21L
  40. If(sides > 3, floor(sides / 2), 2) → sidesL
  41. If(sides > 3, ceil(sides / 2) + 1, 2) → sidesR
  42. Sequence(Segment[Element[pointsR, i], Element[pointsR, i + 1]], i, 0, sidesR) → segmentsR
  43. Sequence(Segment[Element[pointsL, i], Element[pointsL, i + 1]], i, 0, sidesL) → segmentsL
  44. Distance(Element[pointsL, sidesL + 1], Element[pointsR, sidesR + 1]) → openDistanceValue
  45. Rotate(Center + (0, -2), -α, Pt2) → Center2R
  46. Rotate(Center + (0, -2), α, Pt1) → Center2L
  47. Rotate(Center2R, (-open)°, Pt3R) → Center3R
  48. Rotate(Center3R, (-open)°, Pt4R) → Center4R
  49. Rotate(Center4R, (-open)°, Pt5R) → Center5R
  50. Rotate(Center5R, (-open)°, Pt6R) → Center6R
  51. Rotate(Center6R, (-open)°, Pt7R) → Center7R
  52. Rotate(Center7R, (-open)°, Pt8R) → Center8R
  53. Rotate(Center8R, (-open)°, Pt9R) → Center9R
  54. Rotate(Center9R, (-open)°, Pt10R) → Center10R
  55. Rotate(Center10R, (-open)°, Pt11R) → Center11R
  56. Rotate(Center11R, (-open)°, Pt12R) → Center12R
  57. Rotate(Center12R, (-open)°, Pt13R) → Center13R
  58. Rotate(Center13R, (-open)°, Pt14R) → Center14R
  59. Rotate(Center14R, (-open)°, Pt15R) → Center15R
  60. Rotate(Center15R, (-open)°, Pt16R) → Center16R
  61. Rotate(Center16R, (-open)°, Pt17R) → Center17R
  62. Rotate(Center17R, (-open)°, Pt18R) → Center18R
  63. Rotate(Center18R, (-open)°, Pt19R) → Center19R
  64. Rotate(Center19R, (-open)°, Pt20R) → Center20R
  65. Rotate(Center20R, (-open)°, Pt21R) → Center21R
  66. Sequence(Polygon[Element[pointsR, i], Element[pointsR, i + 1], Element[centersR, i]], i, 0, sidesR) → trianglesR
  67. Rotate(Center2L, open°, Pt3L) → Center3L
  68. Rotate(Center3L, open°, Pt4L) → Center4L
  69. Rotate(Center4L, open°, Pt5L) → Center5L
  70. Rotate(Center5L, open°, Pt6L) → Center6L
  71. Rotate(Center6L, open°, Pt7L) → Center7L
  72. Rotate(Center7L, open°, Pt8L) → Center8L
  73. Rotate(Center8L, open°, Pt9L) → Center9L
  74. Rotate(Center9L, open°, Pt10L) → Center10L
  75. Rotate(Center10L, open°, Pt11L) → Center11L
  76. Rotate(Center11L, open°, Pt12L) → Center12L
  77. Rotate(Center12L, open°, Pt13L) → Center13L
  78. Rotate(Center13L, open°, Pt14L) → Center14L
  79. Rotate(Center14L, open°, Pt15L) → Center15L
  80. Rotate(Center15L, open°, Pt16L) → Center16L
  81. Rotate(Center16L, open°, Pt17L) → Center17L
  82. Rotate(Center17L, open°, Pt18L) → Center18L
  83. Rotate(Center18L, open°, Pt19L) → Center19L
  84. Rotate(Center19L, open°, Pt20L) → Center20L
  85. Sequence(Polygon[Element[pointsL, i], Element[pointsL, i + 1], Element[centersL, i - 1]], i, 0, sidesL) → trianglesL
  86. Segment(Center + (0, 0), Center + (0, -1)) → apothemSegment
  87. Segment(Element[pointsR, sidesR + 1] + (0.05, 0), Element[pointsR, sidesR + 1] + (0.55, 0)) → altitudeHashB
  88. Segment(Element[pointsR, sidesR + 1] + (0.05, 1), Element[pointsR, sidesR + 1] + (0.55, 1)) → altitudeHashT
  89. Polygon(Pt1PolyOutline, Pt2PolyOutline, sides) → polygonOutline, a_1, h, i, j, k, l, m, n, p, q, r, pt1polyoutline, s, t, b_1, c_1, d_1, e_1, f_1, g_1, h_1, i_1, a, j_1, k_1, l_1, m_1, n_1, p_1, q_1, r_1, s_1, t_1, b, a_2, A, B, C, D, E, F, G, H, I, c, J, K, L, M, N, O, P, Q, R, S, d, T, U, V, W, Z, A_1, B_1, C_1, D_1, E_1, e, F_1, G_1, H_1, I_1, J_1, K_1, L_1, M_1, N_1, f, g
  90. Vector((x(Center), -0.25), (x(Element[pointsL, sidesL + 1]), -0.25)) → openDistanceVectorL
  91. Vector((x(Center), -0.25), (x(Element[pointsR, sidesR + 1]), -0.25)) → openDistanceVectorR
  92. Vector(Midpoint[Element[pointsR, sidesR + 1] + (0.3, 0), Element[pointsR, sidesR + 1] + (0.3, 1)], Element[pointsR, sidesR + 1] + (0.3, 0)) → altitudeVectorB
  93. Vector(Midpoint[Element[pointsR, sidesR + 1] + (0.3, 0), Element[pointsR, sidesR + 1] + (0.3, 1)], Element[pointsR, sidesR + 1] + (0.3, 1)) → altitudeVectorT
  94. Segment((x(Element[pointsL, sidesL + 1]), -0.1), (x(Element[pointsL, sidesL + 1]), -0.4)) → openDistanceEndLSolid
  95. Segment((x(Element[pointsR, sidesR + 1]), -0.1), (x(Element[pointsR, sidesR + 1]), -0.4)) → openDistanceEndRSolid

AI Drawing Prompt Reference

Draw a regular polygon and show the relationship between the length of the central line segment (from the center to the side) and the perimeter of the polygon. By changing the number of sides, observe how the length of the central line segment affects the perimeter. The relationship can be visualized by rotating and segmenting, starting with the regular polygon, then calculating and displaying the length of the central line segment and the perimeter, and finally emphasizing the symmetry of the polygon and the role of the central line segment through rotations and segmentations.
The prompt above was compiled from the original drawing commands and is for reference only.