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Discovering Area of a Regular Pentagon

Discovering Area of a Regular Pentagon
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2016-03-13 10:44
PolygonTrigonometryGeometric Construction
This example demonstrates how to find the area of a regular polygon (specifically a pentagon) through peeling or dissection. Key math concepts include polygon area formulas, geometric dissection, and angle bisectors. GeoGebra commands and objects used in construction are Polygon, Circle, Center, Segment, Sequence, Mirror, OrthogonalLine, Intersect, Rotate, Angle, and Dilate. Ideal for high school students to understand area calculation methods for complex polygons.

GeoGebra Commands & Parameters

  1. Polygon(A, B, a) → poly1, b, n, p_2, q, r, s, t, a_1, b_1, c_1, d_1, c, e_1, f_1, g_1, h_1, i_1, j_1, k_1, l_1, m_1, n_1, d, p_1, q_1, r_1, s_1, t_1, a_2, b_2, c_2, d_2, e_2, e_3, f_2, C, D, E, F, G, H, I, L, M, f, N, O, P, Q, R, S, T, U, V, W, g, Z, A_1, B_1, C_1, D_1, E_1, F_1, G_1, H_1, I_1, h, J_1, K_1, L_1, M_1, N_1, O_1, P_1, Q_1, R_1, i, j
  2. Circle(A, B, C) → k
  3. Center(k) → J
  4. Segment(J, C) → n_2
  5. Segment(J, B) → m_2
  6. Sequence(Segment[Dilate[A, k, J], Dilate[B, k, J]], k, 0.05, 0.95, 0.05) → list1
  7. Mirror(list1, m_2) → list1'
  8. Mirror(list1', n_2) → list1''
  9. OrthogonalLine(J, b) → l
  10. Mirror(list1'', l) → list1'''
  11. Intersect(Element[list1'', 19], Element[list1''', 19]) → W_1
  12. Intersect(l, b) → K
  13. Segment(J, K) → m
  14. Rotate(K, (360 * ° / a), J) → K'
  15. Angle(K, J, K') → α
  16. Rotate(B, (-α), B) → B'
  17. Intersect(l, poly1) → S_1, T_1
  18. Dilate(J, If[-1 < o ≤ 0, (-o), 0], W_1) → J'
  19. Segment(J, J') → g_2
  20. Dilate(J, If[-2 < o ≤ -1, -1 - o, 0], K) → J'_1
  21. Segment(J, J'_1) → i_2
  22. Angle(B, K, J) → β
  23. Dilate(J, If[-1 < e ≤ 0, (-e), 0], T_1) → J'_2
  24. Dilate(J, If[-2 < e ≤ -1, -1 - e, 0], K) → J'_3
  25. Segment(J, J'_3) → p
  26. Segment(J, J'_2) → j_2
  27. Segment(J, J'_3) → k_2
  28. Segment(J, J'_2) → l_2
  29. Mirror(list1', l) → list1''_1
  30. Rotate(C, If[0 ≤ o ≤ 1, ((-o) * α), (-α)], C) → C'
  31. Rotate(D, If[0 ≤ o ≤ 1, ((-o) * α), (-α)], C) → D'
  32. Segment(C', D') → d'
  33. Mirror(C', l) → C''
  34. Mirror(D', l) → D''
  35. Segment(C'', D'') → d''
  36. Segment(B, D') → q_2
  37. Segment(A, D'') → r_2
  38. Rotate(B, If[1 < o ≤ 2, ((-(o - 1)) * α), (-α)], B) → B'_1
  39. Rotate(D', If[1 < o ≤ 2, ((-(o - 1)) * α), (-α)], B) → D''_1
  40. Segment(B'_1, D''_1) → s_2
  41. Mirror(B'_1, l) → U_1
  42. Mirror(D''_1, l) → V_1
  43. Segment(U_1, V_1) → t_2
  44. Segment(V_1, D''_1) → a_3
  45. Sequence(Intersect[Element[list1', k], Element[list1'', k]], k, 1, 19, 1) → list2
  46. Sequence(Rotate[Element[list1'', k], If[2 < o ≤ 3, ((-(o - 2)) * α), (-α)], Element[list2, k]], k, 1, 19, 1) → list4
  47. Mirror(list4, g_2) → list4'
  48. Sequence(Intersect[Element[list1, k], Element[list1', k]], k, 1, 19, 1) → list6
  49. Sequence(Rotate[Element[list1', k], If[3 < o ≤ 4, ((-(o - 3)) * α), (-α)], Element[list6, k]], k, 1, 19, 1) → list3
  50. Sequence(Rotate[Element[list4, k], If[3 < o ≤ 4, ((-(o - 3)) * α), (-α)], Element[list6, k]], k, 1, 19, 1) → list5
  51. Mirror(list5, g_2) → list5'
  52. Mirror(list3, g_2) → list3'
  53. Dilate(J, If[4 < o ≤ 5, 5 - o, 0], V_1) → J'_4
  54. Segment(J, J'_4) → b_3
  55. Dilate(J, If[4 < o ≤ 5, 5 - o, 0], D''_1) → J'_5
  56. Segment(J, J'_5) → c_3