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Exterior Angle Sum of a Polygon is 360°

Exterior Angle Sum of a Polygon is 360°
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2020-03-16 19:33
PolygonAngleTransformationGeometric ConstructionSlider Animation
① This applet visually demonstrates that the sum of the exterior angles of any polygon (including concave ones) is always 360 degrees. ② The core mathematical concept is the polygon exterior angle sum theorem, involving the definition of exterior angles, rotational direction, and angle accumulation. ③ Construction highlights: uses the Polygon command to create the polygon, Ray and Angle commands to mark exterior angles, Rotate to show how the angles fit together, and Intersect and Circle for auxiliary construction; objects include vertices A, B, C, etc., and multiple angle objects f, g, h, etc. ④ Suitable for middle school to high school students learning geometry theorems, and for teachers to demonstrate an intuitive proof of the exterior angle sum in class.

GeoGebra Commands & Parameters

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