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Area of a Regular Polygon

Area of a Regular Polygon
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2016-03-27 11:07
PolygonVectorTrigonometryGeometric ConstructionSlider Animation
In collections Perimeter & Area Perimeter & Area Area of a triangle and Circle
This example demonstrates how to calculate the area of a regular polygon by dividing it into smaller triangles and using vector operations, rotations, and similarity transformations. Key math concepts include properties of regular polygons, vector operations, rotations, similarity transformations, and trigonometric functions. GeoGebra commands and objects used in the construction include Sequence, Distance, Center, Rotate, Polygon, Vector, Segment, Intersect, Dilate, Translate, and Vertex. It is suitable for high school students learning about area calculation methods for regular polygons and for geometry construction teaching.

GeoGebra Commands & Parameters

  1. Sequence((cos(((k * π) * β / 180)), sin(((k * π) * β / 180))), k, 0, NoSides, 1) → list1
  2. Distance(Element[list1, 1], Element[list1, 2]) → h
  3. Center(c) → A
  4. Sequence(Polygon[Element[list1, k], A, Element[list1, k + 1]], k, 1, NoSides, 1) → list2
  5. Sequence(Vertex[Element[list2, k], 3], k, 2, NoSides, 1) → list4
  6. Sequence(Vertex[Element[list2, k], 3], k, 1, NoSides - 1) → list5
  7. Rotate(A, (((-b) * 360) * ° / NoSides), B_1) → A'
  8. Sequence(Rotate[Element[list1, k], ((((-(k - 1)) * b) * 360) * ° / NoSides), Element[list1, k - 1]], k, 2, NoSides, 1) → list3
  9. Sequence(Rotate[Element[list3, k], ((((-(k - 1)) * b) * 360) * ° / NoSides), Element[list3, k - 1]], k, 2, NoSides - 1, 1) → list6
  10. Polygon(Element[list1, 1], A', Element[list3, 1]) → poly1, d, e, a'
  11. Vector(Element[list1, 2], Element[list3, 1]) → u
  12. Sequence(Vector[Element[list1, k], Element[list3, k - 1]], k, 2, NoSides, 1) → list7
  13. Segment((0, -1), (6.283185307179586, -1)) → f
  14. Intersect(c, d, 1) → D
  15. Segment(D, C) → i
  16. Dilate(D, If[0 < g ≤ 1, 1 - g, 0], C) → D'
  17. Vector(D, D') → v
  18. Translate(list1, v) → list1'
  19. Translate(c, v) → c'
  20. Translate(list2, v) → list2'
  21. Rotate(Element[list2', 1], (((-g) * 360 / NoSides))°, Element[list1', 1]) → poly2
  22. Vertex(Element[list2', 2], 1) → E
  23. Vertex(poly2, 3) → F
  24. Vector(Vertex[Element[list2', 2], 1], Vertex[poly2, 3]) → w
  25. Translate(Element[list2', 2], w) → poly3
  26. Rotate(poly3, ((((-g) * 2) * 360 / NoSides))°, F) → poly3'
  27. Vertex(poly3', 3) → G
  28. Vector(Vertex[Element[list2', 3], 1], Vertex[poly3', 3]) → j
  29. Translate(Element[list2', 3], j) → poly4
  30. Rotate(poly4, ((((-g) * 3) * 360 / NoSides))°, G) → poly4'
  31. Vertex(poly4', 3) → H
  32. Vector(Element[list1', 4], H) → k
  33. Translate(Element[list2', 4], k) → poly5
  34. Rotate(poly5, ((((-g) * 4) * 360 / NoSides))°, H) → poly5'
  35. Vertex(poly5', 3) → I
  36. Vector(Element[list1', 5], I) → l
  37. Translate(Element[list2', 5], l) → poly6
  38. Rotate(poly6, ((((-g) * 5) * 360 / NoSides))°, I) → poly6'
  39. Vertex(poly6', 3) → J
  40. Vector(Element[list1', 6], J) → m
  41. Translate(Element[list2', 6], m) → poly7
  42. Rotate(poly7, ((((-g) * 6) * 360 / NoSides))°, J) → poly7'
  43. Rotate(poly7, ((((-g) * 6) * 360 / NoSides))°, J) → poly7'_1
  44. Vertex(poly7'_1, 3) → K
  45. Vector(Element[list1', 7], K) → n
  46. Translate(Element[list2', 7], n) → poly8
  47. Rotate(poly8, ((((-g) * 7) * 360 / NoSides))°, K) → poly8'
  48. Vertex(poly8', 3) → L
  49. Vector(Element[list1', 8], L) → p
  50. Translate(Element[list2', 8], p) → poly9
  51. Rotate(poly9, ((((-g) * 8) * 360 / NoSides))°, L) → poly9'
  52. Vertex(poly9', 3) → M
  53. Vector(Element[list1', 9], M) → q
  54. Translate(Element[list2', 9], q) → poly10
  55. Vertex(poly2, 2) → B
  56. Vertex(poly3', 2) → N
  57. Vertex(poly4', 2) → O
  58. Vertex(poly5', 2) → P
  59. Vertex(poly6', 2) → Q
  60. Vertex(poly7'_1, 2) → R
  61. Vertex(poly8', 2) → S
  62. Vertex(poly9', 2) → T
  63. Rotate(poly2, (((-o) * 180))°, B) → poly2'
  64. Rotate(poly3', (((-o) * 180))°, N) → poly3''
  65. Rotate(poly4', (((-o) * 180))°, O) → poly4''
  66. Rotate(poly5', (((-o) * 180))°, P) → poly5''
  67. Rotate(poly6', (((-o) * 180))°, Q) → poly6''
  68. Rotate(poly7'_1, (((-o) * 180))°, R) → poly11
  69. Rotate(poly8', (((-o) * 180))°, S) → poly8''
  70. Rotate(poly9', (((-o) * 180))°, T) → poly9''
  71. Dilate(B, 1 - r, C) → B'
  72. Dilate(N, 1 - r, F) → N'
  73. Dilate(O, 1 - r, G) → O'
  74. Dilate(P, 1 - r, H) → P'
  75. Dilate(Q, 1 - r, I) → Q'
  76. Dilate(R, 1 - r, J) → R'
  77. Dilate(S, 1 - r, K) → S'
  78. Dilate(T, 1 - r, L) → T'
  79. Vector(N, N') → s
  80. Vector(B, B') → t
  81. Vector(O, O') → u_1
  82. Vector(P, P') → v_1
  83. Vector(Q, Q') → w_1
  84. Vector(R, R') → a_1
  85. Vector(S, S') → b_1
  86. Vector(T, T') → c_1
  87. Translate(poly2', t) → poly2''
  88. Translate(poly3'', s) → poly3'''
  89. Translate(poly4'', u_1) → poly4'''
  90. Translate(poly5'', v_1) → poly5'''
  91. Translate(poly6'', w_1) → poly6'''
  92. Translate(poly11, a_1) → poly11'
  93. Translate(poly8'', b_1) → poly8'''
  94. Translate(poly9'', c_1) → poly9'''

AI Drawing Prompt Reference

Draw a regular polygon and divide it into several triangles to calculate its area using rotations and similarity transformations. Start by creating a circle c with center A and radius 1, then generate a list of vertices list1 for the regular polygon. Calculate the side length h, and construct the center A' and triangle list2. Gradually build each part of the polygon through rotations and similarity transformations, and finally obtain the area of the regular polygon.
The prompt above was compiled from the original drawing commands and is for reference only.