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Sum of Interior Angles of Any Polygon: Explore & Verify

Sum of Interior Angles of Any Polygon: Explore & Verify
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2023-09-11 18:20
PolygonAngleSequenceTransformationGeometric Construction
This GeoGebra example demonstrates how the sum of interior angles changes as a polygon’s number of sides increases, and it supports guided reasoning by having students predict the answers for a triangle, a 10-gon, and a 100-gon. Key concepts include polygon interior angles, the interior-angle-sum relationship, dividing a polygon into triangles as a reasoning strategy, and measuring/aggregating angles to verify the pattern. Construction highlights: Polygon builds the polygon from vertices; Point, Segment, and Angle compute each interior angle; geometric transformations such as Rotate, Dilate, and Mirror help position or replicate components needed for the construction workflow; Intersect plus Circle/CircleSector provide auxiliary construction geometry (e.g., reference points or angle regions); Midpoint and If support automation for selecting/organizing objects. The angles are collected (e.g., allAng4/5/6) and displayed with question/result text (textQuestion, textAllIn). Suitable for Grades 6–8 as a classroom activity, investigation, or verification of the interior-angle-sum rule.

GeoGebra Commands & Parameters

  1. Mirror(P1, yAxis) → P1'
  2. Mirror(P1', xAxis) → P1''
  3. Mirror(P1'', yAxis) → P1'''
  4. Polygon(P1, P1', P1'', P1''') → peep, p1, p1', p1'', p1'''
  5. SelectedIndex(ddMenu) → is
  6. Mirror(A, x = x(CEN)) → B
  7. Intersect(xAxis, yAxis) → O
  8. Circle(CEN, A) → c
  9. Point(c) → C
  10. Point(c) → D
  11. Point(c) → E
  12. Point(c) → F
  13. Polygon(If[is ≟ 1, {A, B, C, D}, is ≟ 2, {A, B, C, D, E}, {A, B, C, D, E, F}]) → poli
  14. Dilate(B, If[0 < aniSli ≤ 1, 1 - aniSli, 0], D) → B'
  15. Segment(B, B') → segmBD
  16. Angle(A, B, D) → allAng4
  17. Angle(D, A, B) → allAng5
  18. Angle(B, D, A) → allAng6
  19. Angle(D, B, C) → allAng1
  20. Angle(C, D, B) → allAng2
  21. Angle(B, C, D) → allAng3
  22. Dilate(A, If[0 < aniSli ≤ 1, 1 - aniSli, 0], D) → A'
  23. Segment(A, A') → segmAD
  24. Angle(E, A, D) → fiveAng1
  25. Angle(D, E, A) → fiveAng2
  26. Angle(A, D, E) → fiveAng3
  27. Dilate(F, If[0 < aniSli ≤ 1, 1 - aniSli, 0], D) → F'
  28. Segment(F, F') → segmFD
  29. Angle(F, A, D) → sixAng1
  30. Angle(D, F, A) → sixAng2
  31. Angle(A, D, F) → sixAng3
  32. Angle(E, F, D) → sixAng4
  33. Angle(D, E, F) → sixAng5
  34. Angle(F, D, E) → sixAng6
  35. Segment(G, H) → angBase3
  36. Segment(K, L) → angBase2
  37. Midpoint(I, J) → M1
  38. Midpoint(K, L) → M2
  39. Midpoint(G, H) → M3
  40. Segment(I, J) → angBase1
  41. Rotate(J, allAng1, M1) → J1
  42. Rotate(J1, allAng2, M1) → J2
  43. CircleSector(M1, J, J1) → allAng1Recap
  44. CircleSector(M1, J1, J2) → allAng2Recap
  45. CircleSector(M1, J2, I) → allAng3Recap
  46. Rotate(L, allAng4, M2) → L1
  47. Rotate(L1, allAng5, M2) → L2
  48. CircleSector(M2, L, L1) → allAng4Recap
  49. CircleSector(M2, L1, L2) → allAng5Recap
  50. CircleSector(M2, L2, K) → allAng6Recap
  51. Rotate(H, fiveAng1, M3) → H1
  52. Rotate(H1, fiveAng2, M3) → H2
  53. CircleSector(M3, H, H1) → fiveAng1Recap
  54. CircleSector(M3, H1, H2) → fiveAng2Recap
  55. CircleSector(M3, H2, G) → fiveAng3Recap
  56. If(is ≟ 2, H + (3, 0), L + (3, 0)) → M
  57. Midpoint(M, N) → M4
  58. Segment(M, N) → angBase4
  59. Rotate(N, sixAng1, M4) → N1
  60. Rotate(N1, sixAng2, M4) → N2
  61. CircleSector(M4, N, N1) → sixAng1Recap
  62. CircleSector(M4, N1, N2) → sixAng2Recap
  63. CircleSector(M4, N2, M) → sixAng3Recap
  64. Segment(P, Q) → angBase5
  65. Midpoint(P, Q) → M5
  66. Rotate(Q, sixAng4, M5) → Q1
  67. Rotate(Q1, sixAng5, M5) → Q2
  68. CircleSector(M5, Q, Q1) → sixAng4Recap
  69. CircleSector(M5, Q1, Q2) → sixAng5Recap
  70. CircleSector(M5, Q2, P) → sixAng6Recap
  71. Segment(M1, J1 + (0.5 * UnitVector[Segment[M1, J1]])) → div11
  72. Segment(M1, J2 + (0.5 * UnitVector[Segment[M1, J2]])) → div12
  73. Segment(M2, L1 + (0.5 * UnitVector[Segment[M2, L1]])) → div21
  74. Segment(M2, L2 + (0.5 * UnitVector[Segment[M2, L2]])) → div22
  75. Segment(M3, H1 + (0.5 * UnitVector[Segment[M3, H1]])) → div31
  76. Segment(M3, H2 + (0.5 * UnitVector[Segment[M3, H2]])) → div32
  77. Segment(M4, N1 + (0.5 * UnitVector[Segment[M4, N1]])) → div41
  78. Segment(M4, N2 + (0.5 * UnitVector[Segment[M4, N2]])) → div42
  79. Segment(M5, Q1 + (0.5 * UnitVector[Segment[M5, Q1]])) → div51
  80. Segment(M5, Q2 + (0.5 * UnitVector[Segment[M5, Q2]])) → div52
  81. circlePixels(C, 7 + (7 * pulsing), Corner[1], Corner[3], Corner[5]) → pulsingObject

AI Drawing Prompt Reference

Draw a diagram to verify the sum of interior angles of a polygon. Start with a square as reference, then a circle with several points on it. Based on a dropdown selection, construct a polygon with three, four, or five vertices from these points. Connect one vertex to others to divide the polygon into triangles, and label each interior angle. Use circular arcs to represent each angle measure, and display the total sum. Include an animation slider that controls auxiliary segments from a vertex to opposite sides, illustrating how the polygon is triangulated to derive the angle sum.
The prompt above was compiled from the original drawing commands and is for reference only.