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Perimeter Demo: Circumference, Square, and Arbitrary Triangle

Perimeter Demo: Circumference, Square, and Arbitrary Triangle
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2014-06-25 00:05
TriangleCirclePolygonGeometric Construction
① This interactive GeoGebra applet visually demonstrates the concept of perimeter by displaying and comparing the perimeters of a circle (circumference), a square, and an arbitrary triangle. ② Core mathematical concepts include the definition of perimeter, the circumference formula (2πr), square perimeter (4a), triangle perimeter as sum of three sides, with connections to the Pythagorean theorem for side length calculations and linear regression for possible data fitting. ③ Construction highlights: Based on the construction summary, the applet uses GeoGebra commands such as Segment, Circle, CircleSector, CircleArc, Distance, Angle, Rotate, Intersect, OrthogonalLine, Line, and Ray. It defines multiple objects including points (A, B, M, N, J, K), segments (a, b, d_1, lsquare, ltriangle1/2), circles (c_1, e_1), and parameter objects (triangleparam, sumtriangle, phasesquare, go) to enable dynamic step-by-step visualization via sliders. ④ Suitable for upper elementary to middle school students learning basic geometry, and for teachers to illustrate perimeter formulas and comparisons in an interactive classroom setting.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. OrthogonalLine(A, a) → b
  3. Distance(A, B) → Pmt
  4. Circle(A, Pmt / 2) → d_1
  5. Intersect(d_1, b, 2) → L
  6. Rotate(L, 89°, A) → L'
  7. CircleSector(A, L, L') → e_1
  8. PointIn(e_1) → T1
  9. Distance(A, T1) → ltriangle2
  10. Circle(A, ltriangle2) → p_1
  11. Intersect(p_1, a, 1) → N
  12. Angle(N, A, T1) → RT1
  13. Ellipse(A, T1, triangleparam / 2) → f_1
  14. Point(f_1) → T2
  15. If(alphatrianglestart ≟ (Angle[B, A, T2] * 360 / 2 / π), 1, 2) → casetriangle
  16. If(go < alphatrianglestart / sumtriangle, 1, If[go < (alphatrianglestart + 180 - angleT2) / sumtriangle, 2, 3]) → phasetriangle1
  17. If(casetriangle ≟ 2 ∧ phasetriangle1 ≟ 1, RT1 - ((go / alphatrianglestart * sumtriangle) * RT1)) → alpha2triangle1
  18. Distance(T2, A) → ltriangle1
  19. Circle(A, ltriangle1) → k_1
  20. Intersect(k_1, a, 1) → M
  21. Angle(M, A, T2) → RT2
  22. If(casetriangle ≟ 1 ∧ phasetriangle1 ≟ 1, RT2 - ((go / alphatrianglestart * sumtriangle) * RT2)) → alpha1triangle1
  23. Circle(A, lsquare) → g
  24. Intersect(g, b, 2) → J
  25. Rotate(J, 89°, A) → J'
  26. CircleArc(A, J, J') → c_1
  27. Point(c_1) → K
  28. Angle(J, A, K) → alphasquarestart
  29. If(go < alphasquarestart / sum, 1, If[go < (alphasquarestart + 90°) / sum, 2, If[go < (alphasquarestart + 180°) / sum, 3, 4]]) → phasesquare
  30. Intersect(g, a, 1) → H
  31. Circle(H, lsquare) → c
  32. If(phasesquare ≟ 1, 90° + (alphasquarestart * (1 - (go * sum / alphasquarestart)))) → alphasquare1
  33. Intersect(c, a, 1) → I
  34. Line(D, b) → e
  35. Distance(A, D) → parc
  36. Circle(A, rcircle) → ccircle
  37. Intersect(ccircle, b, 2) → C
  38. Line(C, a) → f
  39. Intersect(e, f) → E
  40. Rotate(D, alphacircle, E) → D'
  41. Angle(D, E, D') → α
  42. CircleArc(E, D, D') → gcircle
  43. Segment(A, D) → hcircle
  44. Rotate(H, alphasquare1, A) → G
  45. Rotate(G, -90°, A) → G'
  46. Rotate(A, 90°, G) → F'
  47. Segment(F', G') → i
  48. Segment(F', G) → j
  49. Segment(G, A) → d
  50. Segment(A, G') → h
  51. If(phasesquare ≟ 2, 90° + ((90 * (1 - ((go - alphasquarestart / sum) * sum / (90°)))))°) → alphasquare2
  52. Rotate(I, alphasquare2, H) → I'
  53. Angle(I, H, I') → β
  54. Rotate(H, 90°, I') → H'
  55. Angle(H, I', H') → γ
  56. Rotate(I', 90°, H') → I''
  57. Angle(I', H', I'') → δ
  58. Segment(I', H') → k
  59. Segment(H', I'') → l
  60. Segment(I'', H) → m
  61. Segment(H, A) → n
  62. Circle(I, lsquare) → p
  63. Intersect(p, a, 2) → F
  64. If(phasesquare ≟ 3, ((90 * (1 - ((go - (alphasquarestart + 90°) / sum) * sum / (90°)))))°) → alphasquare3
  65. Rotate(F, alphasquare3, I) → F'_1
  66. Angle(F, I, F'_1) → ε
  67. Rotate(I, -90°, F'_1) → I'_1
  68. Angle(I'_1, F'_1, I) → ζ
  69. Segment(I'_1, F'_1) → q
  70. Segment(F'_1, I) → r
  71. If(phasesquare ≟ 4, ((90 * (1 - ((go - (alphasquarestart + 180°) / sum) * sum / (90°)))))°) → alphasquare4
  72. Rotate(B, alphasquare4, F) → B'
  73. Angle(B, F, B') → η
  74. Segment(B', F) → s
  75. Segment(H, I) → t
  76. Segment(I, F) → a_1
  77. Angle(J, A, J') → θ
  78. Angle(L, A, L') → ι
  79. Distance(T1, T2) → ltriangleA
  80. Rotate(M, alpha1triangle1, A) → M'
  81. Angle(M, A, M') → κ
  82. Rotate(N, alpha2triangle1, A) → N'
  83. Angle(N, A, N') → λ
  84. If(go < alphatrianglestart / sumtriangle, 1, If[go < (alphatrianglestart + 180 - angleT1) / sumtriangle, 2, 3]) → phasetriangle2
  85. Segment(M', A) → h_1
  86. Circle(M', ltriangleA) → q_1
  87. Ray(M', A) → i_1
  88. Intersect(q_1, i_1, 1) → O
  89. Rotate(O, (-(angleT2°)), M') → O'
  90. Angle(O', M', O) → μ
  91. Segment(O', M') → j_1
  92. Segment(O', A) → l_1
  93. Segment(N', A) → b_1
  94. Circle(N', ltriangleA) → g_1
  95. Ray(N', A) → m_1
  96. Intersect(g_1, m_1, 1) → P
  97. Rotate(P, (-(angleT1°)), N') → P'
  98. Angle(P', N', P) → ν
  99. Segment(N', P') → n_1
  100. Segment(P', A) → r_1
  101. Circle(M, ltriangleA) → s_1
  102. Intersect(s_1, a, 1) → Q
  103. If(casetriangle ≟ 1 ∧ phasetriangle1 ≟ 2, (((180 - angleT2) * (1 - ((go - alphatrianglestart / sumtriangle) / (180 - angleT2) * sumtriangle))))°) → alpha1triangle2
  104. Rotate(Q, alpha1triangle2, M) → Q'
  105. Angle(Q, M, Q') → ξ
  106. Segment(Q', M) → t_1
  107. Circle(Q', ltriangle2) → c_2
  108. Ray(Q', M) → a_2
  109. Intersect(c_2, a_2, 1) → R
  110. Rotate(R, (-(angleT1°)), Q') → R'
  111. Angle(R', Q', R) → ο
  112. Segment(Q', R') → b_2
  113. If(casetriangle ≟ 1 ∧ phasetriangle1 ≟ 3, (((180 - angleT1) * (1 - ((go - (alphatrianglestart + 180 - angleT2) / sumtriangle) / (180 - angleT1) * sumtriangle))))°) → alpha1triangle3
  114. Rotate(B, alpha1triangle3, Q) → B'_1
  115. Angle(B, Q, B'_1) → ρ
  116. Segment(B'_1, Q) → e_2
  117. If(casetriangle ≟ 2 ∧ phasetriangle2 ≟ 2, (((180 - angleT1) * (1 - ((go - alphatrianglestart / sumtriangle) / (180 - angleT1) * sumtriangle))))°) → alpha2triangle2
  118. Circle(N, ltriangleA) → d_2
  119. Intersect(d_2, a, 2) → S
  120. Rotate(S, alpha2triangle2, N) → S'
  121. Angle(S, N, S') → σ
  122. Circle(S', ltriangle1) → f_2
  123. Ray(S', N) → g_2
  124. Intersect(f_2, g_2, 1) → T
  125. Rotate(T, (-(angleT2°)), S') → T'
  126. Angle(T', S', T) → τ
  127. Segment(S', T') → h_2
  128. Segment(N, S') → i_2
  129. Circle(S, ltriangle1) → k_2
  130. If(casetriangle ≟ 2 ∧ phasetriangle2 ≟ 3, (((180 - angleT2) * (1 - ((go - (alphatrianglestart + 180 - angleT1) / sumtriangle) / (180 - angleT2) * sumtriangle))))°) → alpha2triangle3
  131. Rotate(B, alpha2triangle3, S) → B'_2
  132. Angle(B, S, B'_2) → υ
  133. Segment(B'_2, S) → j_2
  134. Segment(A, M) → l_2
  135. Segment(M, Q) → m_2
  136. Segment(A, N) → n_2
  137. Segment(N, S) → p_2

AI Drawing Prompt Reference

Create a diagram comparing the perimeters of a circle, a square, and an arbitrary triangle. Start with a horizontal base segment, use it as the radius to draw a circle, and place a point on the circle to form a sector, illustrating the circumference. Then construct a square by rotating a point and finding intersections, showing its side length and perimeter. Finally, construct an arbitrary triangle using an ellipse and angle parameters to adjust its shape, and compute the sum of its three sides. All shapes share a common center, with sliders controlling the expansion of each shape to visually compare their perimeters.
The prompt above was compiled from the original drawing commands and is for reference only.