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Investigation: Area Formula for Regular Polygons

Investigation: Area Formula for Regular Polygons
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2015-12-12 13:06
PolygonAngleTrigonometryGeometric Construction
This example demonstrates how to discover the area formula for regular n-gons by examining regular quadrilaterals, pentagons, and hexagons. It involves core mathematical concepts such as polygons, area calculation, geometric construction, and trigonometry. The construction uses GeoGebra commands and objects like Polygon, Segment, Intersect, OrthogonalLine, Angle, Rotate, and Ray. It is suitable for high school students or math enthusiasts exploring the derivation of area formulas for regular polygons.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  2. Segment(D, B) → e
  3. Segment(A, C) → f
  4. Intersect(e, f) → E
  5. OrthogonalLine(E, a) → g
  6. Intersect(g, a) → F
  7. Segment(E, F) → h
  8. Segment(A, E) → i
  9. Segment(A, F) → j
  10. Polygon(G, H, 5) → poly2, k, l, m, n, p, I, J, K
  11. Angle(H, G, K) → α
  12. Angle(G, K, J) → β
  13. Angle(K, J, I) → γ
  14. Angle(J, I, H) → δ
  15. Angle(I, H, G) → ε
  16. Rotate(H, ζ, G) → H'
  17. Angle(H, G, H') → η
  18. Ray(G, H') → q
  19. Rotate(G, ζ, K) → G'
  20. Angle(G, K, G') → θ
  21. Rotate(K, ζ, J) → K'
  22. Angle(K, J, K') → ι
  23. Rotate(J, ζ, I) → J'
  24. Angle(J, I, J') → κ
  25. Rotate(I, ζ, H) → I'
  26. Angle(I, H, I') → λ
  27. Ray(K, G') → r
  28. Ray(J, K') → s
  29. Ray(I, J') → t
  30. Ray(H, I') → a_1
  31. Intersect(q, r) → L
  32. OrthogonalLine(L, k) → b_1
  33. Segment(G, L) → c_1
  34. Intersect(b_1, k) → M
  35. Segment(L, M) → d_1
  36. Polygon(N, O, 6) → poly3, e_1, S, f_1, g_1, h_1, i_1, j_1, P, Q, R
  37. Segment(E, B) → k_1
  38. Segment(L, H) → l_1
  39. Angle(O, N, S) → μ
  40. Angle(N, S, R) → ν
  41. Rotate(O, ξ, N) → O'
  42. Angle(O, N, O') → ο
  43. Ray(N, O') → m_1
  44. Rotate(N, ξ, S) → N'
  45. Angle(N, S, N') → ρ
  46. Ray(S, N') → n_1
  47. Intersect(m_1, n_1) → T
  48. Segment(N, T) → p_1
  49. Segment(T, O) → q_1
  50. OrthogonalLine(T, e_1) → r_1
  51. Intersect(r_1, e_1) → U
  52. Segment(U, T) → s_1