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Area of a Parallelogram Formula

Area of a Parallelogram Formula
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2024-02-06 18:07
PolygonGeometric ConstructionSlider Animation
In collections Parallelogramm und Raute: Flächeninhalt
This example guides students through three progressive open questions exploring the parallelogram area formula. It demonstrates: (1) calculating area given base a=6cm and corresponding height ha=2cm to get A=12cm²; (2) the core concept that area = base × corresponding height, understanding why different side-height combinations yield identical results; (3) construction details: using Polygon for the parallelogram, Point/Segment for vertices and edges, OrthogonalLine for heights, Intersect for foot of perpendicular, and Translate to transform into a rectangle; (4) target audience: students in grades 5-6 learning area formulas, developing geometric intuition and reasoning skills.

GeoGebra Commands & Parameters

  1. Textfield(input1) → InputBox1
  2. Textfield(input2) → InputBox2
  3. Textfield(input3) → InputBox3
  4. Textfield(input4) → InputBox4
  5. SpeechBubbleRight(E, 10, 2) → Vieleck1
  6. Polygon(F, H, G, I) → v1, f, g, h, i
  7. Spline({F, Midpoint[F, H], H}, 3) → AnimStrecke
  8. SelectedIndex(DD1) → index1
  9. SelectedIndex(DD2) → index2
  10. SelectedIndex(DD3) → index3
  11. Segment(TriAnimP2, TriAnimP3) → j
  12. Angle(TriAnimP3, TriAnimP2, TriAnimP1) → α
  13. Segment(TriAnimP3, TriAnimP1) → k
  14. Segment(TriAnimP2, TriAnimP1) → l
  15. SpeechBubbleRight(E, 9, 1.8) → Vieleck2
  16. Segment(I, J) → m
  17. Segment(J, H) → n
  18. Textfield(InputText) → InputBox5
  19. Polygon(I, J, F) → v2, p, q, r
  20. Polygon(TriAnimP1, TriAnimP2, TriAnimP3) → d1, trianimp3, trianimp1, trianimp2
  21. Polygon(I, J, H, G) → v3, i_2, j_1, h_2, g_2
  22. SelectedIndex(DD4) → index4
  23. SelectedIndex(DD5) → index5
  24. Segment(G, H) → Slide2
  25. Point(Slide2) → DynP2
  26. OrthogonalLine(DynP2, g) → s
  27. Intersect(s, Segment[F, I]) → O
  28. Point(Segment[DynP2, Anim2EndP], schieb3) → P
  29. Polygon(P, P + Vector[DynP2, O], P + Vector[DynP2, F], P + Vector[DynP2, H]) → v4, p_1, t, a, b
  30. Polygon(DynP2, O, I, G) → v5, dynp2, o, i_1, g_1
  31. SpeechBubbleRight(POSSpeechStep6, 10, 1.6) → SpeechStep6
  32. Angle(P + Vector[DynP2, F], P + Vector[DynP2, O], P) → β
  33. Text("$\textsf{\textcolor{db6114}{\bold{h_b}}}$", Midpoint[P, P + Vector[DynP2, O]] - (0.5 * UnitOrthogonalVector[Segment[DynP2, O]]), false, true, 0, 0) → Text4
  34. SpeechBubbleRight(R, 11.5, 2) → SpeechBubbleStep7
  35. Translate(v4, Vector[F, POSVEnd2]) → v4'
  36. Translate(DynP2, Vector[F, POSVEnd2]) → DynP2'
  37. Translate(O, Vector[F, POSVEnd2]) → O'
  38. Translate(I, Vector[F, POSVEnd2]) → I'
  39. Translate(G, Vector[F, POSVEnd2]) → G'
  40. Polygon(DynP2', O', I', G') → v5', dynp2', o', i', g'
  41. Translate(TriAnimP1, Vector[F, POSVEnd1]) → TriAnimP1'
  42. Translate(TriAnimP2, Vector[F, POSVEnd1]) → TriAnimP2'
  43. Translate(TriAnimP3, Vector[F, POSVEnd1]) → TriAnimP3'
  44. Polygon(TriAnimP1', TriAnimP2', TriAnimP3') → d1', trianimp3', trianimp1', trianimp2'
  45. Translate(I, Vector[F, POSVEnd1]) → I'_{1}
  46. Translate(J, Vector[F, POSVEnd1]) → J'
  47. Translate(H, Vector[F, POSVEnd1]) → H'
  48. Translate(G, Vector[F, POSVEnd1]) → G'_{1}
  49. Polygon(I'_{1}, J', H', G'_{1}) → v3', i'_{1}, j', h', g'_{1}
  50. Angle(Translate[P + Vector[DynP2, F], Vector[F, POSVEnd2]], Translate[P + Vector[DynP2, O], Vector[F, POSVEnd2]], Translate[P, Vector[F, POSVEnd2]]) → γ
  51. Angle(TriAnimP3', TriAnimP2', TriAnimP1') → δ
  52. PolyLine(G'_{1}, H', J', I'_{1}, G'_{1}) → VEndPoly1
  53. PolyLine(TriAnimP3', TriAnimP2', TriAnimP1', TriAnimP3') → VEndPoly2
  54. Translate(a, Vector[F, POSVEnd2]) → a'
  55. Translate(b, Vector[F, POSVEnd2]) → b'
  56. Translate(p_1, Vector[F, POSVEnd2]) → c
  57. Translate(t, Vector[F, POSVEnd2]) → t'
  58. Segment(DynP2, P) → d
  59. Translate(F, Vector[F, EndSkizz3POS]) → EndSkizz3P1
  60. Translate(H, Vector[F, EndSkizz3POS]) → EndSkizz3P2
  61. Translate(G, Vector[F, EndSkizz3POS]) → EndSkizz3P3
  62. Translate(I, Vector[F, EndSkizz3POS]) → EndSkizz3P4
  63. Midpoint(EndSkizz3P1, EndSkizz3P3) → EndSkizz3M
  64. PolyLine({EndSkizz3M + (Vector[EndSkizz3M, EndSkizz3P1] * relsizeEndSkizz), EndSkizz3M + (Vector[EndSkizz3M, EndSkizz3P2] * relsizeEndSkizz), EndSkizz3M + (Vector[EndSkizz3M, EndSkizz3P3] * relsizeEndSkizz), EndSkizz3M + (Vector[EndSkizz3M, EndSkizz3P4] * relsizeEndSkizz), EndSkizz3M + (Vector[EndSkizz3M, EndSkizz3P1] * relsizeEndSkizz)}) → EndSkizz3Outline
  65. Point(EndSkizz3Outline) → T
  66. Point(EndSkizz3Outline) → U
  67. Point(EndSkizz3Outline) → V
  68. Point(EndSkizz3Outline) → W
  69. Segment(W, V) → e
  70. Intersect(OrthogonalLine[U, e], e) → Z
  71. Point(EndSkizz3Outline) → A_1
  72. Intersect(OrthogonalLine[T, Segment[A_1, W]], Segment[A_1, W]) → B_1
  73. Segment(B_1, T) → radcon_1
  74. Segment(Z, U) → radcon_2
  75. Translate(TriAnimP2, Vector[F, POSVEnd1]) → TriAnimP2'_{1}
  76. Translate(TriAnimP3, Vector[F, POSVEnd1]) → TriAnimP3'_{1}
  77. Segment(TriAnimP2'_{1}, TriAnimP3'_{1}) → j'_{1}
  78. Translate(p_1, Vector[F, POSVEnd2]) → f_1

AI Drawing Prompt Reference

Draw a parallelogram with base a=6cm and height ha=2cm, calculate the area A=12cm². Use the polygon tool to draw the parallelogram, define vertices and edges with points and segments, use perpendicular lines to define heights, and find the foot of the perpendicular through intersection. Transform the parallelogram into a rectangle through translation to show that different side-height combinations yield the same area.
The prompt above was compiled from the original drawing commands and is for reference only.