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Area of Basic Shapes: Parallelogram, Trapezoid, and Quadratic Functions

Area of Basic Shapes: Parallelogram, Trapezoid, and Quadratic Functions
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2018-05-25 17:40
PolygonAngleFunctionDerivativeTransformation
① This GeoGebra example dynamically demonstrates the area calculation of basic plane shapes such as parallelograms and trapezoids, while integrating quadratic functions, derivatives, step functions, and volume concepts to create a multi-level mathematical exploration environment. ② Core mathematical concepts include area formulas for parallelograms and trapezoids, properties of quadratic functions, derivatives (slope changes), step functions (piecewise areas), and volumes formed by rotation or translation. ③ Construction highlights: using commands like Line, Point, Intersect, Polygon, Angle, Translate, and Rotate to build base shapes; sliders (e.g., stepspara) control parameters; multiple polygons (poly1, poly2) and their transformed copies (poly1', poly2') are generated, with angle α labeled to show relationships between area and parameters. ④ Suitable for middle and high school mathematics teaching, providing intuitive understanding of geometric area, functions, and introductory calculus, and can be used as an interactive demonstration or student exploration tool.

GeoGebra Commands & Parameters

  1. Line(B, C) → f
  2. Line(B, A) → g
  3. Line(A, f) → h
  4. Line(C, g) → i
  5. Point(h) → D
  6. Intersect(h, i) → E
  7. Polygon(B, A, D, C) → q1, b, a, d, c
  8. Polygon(B, A, E, C) → q2, b_1, a_1, e, c_1
  9. Angle(C, B, A) → α
  10. Polygon(C, B + ((E - C) * c / ((b * abs(cos(α))))), A, A + ((C - B) * (2 - ((b * cos(α))) / c)), C + ((E - C) * c / ((b * cos(α))))) → poly1, k, l, m, n, p
  11. Polygon(E, A + ((C - B) * (2 - ((b * cos(α))) / c)), C + ((E - C) * c / ((b * cos(α))))) → poly2, q, r, s
  12. Translate(poly1, ((B - C) * Min[(j * stepspara), 1])) → poly1'
  13. Translate(poly2, (((B - C) * j) * stepspara)) → poly2'
  14. Polygon(C, E, A + ((E - A) * (1 - (b * abs(cos(α)) / c)))) → t1, t, c_2, e_1
  15. Translate(t1, ((B - C) * j)) → t1'
  16. Polygon(A, B, C, A + ((E - A) * (1 - (b * abs(cos(α)) / c)))) → t2, f_1, g_1, h_1, i_1
  17. Polygon(C, B + ((E - C) * c / ((b * abs(cos(α))))), B) → poly3, j_1, k_1, l_1
  18. Polygon(B, C, A) → t3, a_2, b_2, c_3
  19. Rotate(B, ((j * 180))°, F) → B'
  20. Rotate(C, ((j * 180))°, F) → C'
  21. Rotate(A, ((j * 180))°, F) → A'
  22. Polygon(B', C', A') → t3', a', b', c'
  23. Rotate(B, (180° * j), F) → B'_1
  24. Rotate(A, (180° * j), F) → A'_1
  25. Rotate(D, (180° * j), F) → D'
  26. Rotate(C, (180° * j), F) → C'_1
  27. Polygon(B'_1, A'_1, D', C'_1) → q1', b'_1, a'_1, d', c'_1