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Hexagon Cross Sections Volume

Hexagon Cross Sections Volume
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2018-03-19 01:54
Solid GeometryCalculusParametric EquationGeometric Construction
In collections Volumes in Calculus Volumes in Calculus
This example demonstrates the calculation of the volume formed by hexagon cross sections with a diagonal. It involves core math concepts like volume, continuity, equilateral triangles, parametric curves, and median value. The construction uses commands like Polygon, Surface, Segment, Sequence, and Integral, suitable for high school and above in calculus education.

GeoGebra Commands & Parameters

  1. Polygon((s, g(s) + (f(s) - g(s)) / 4, ((-(f(s) - g(s))) * sqrt(3) / 4)), (s, f(s) - (f(s) - g(s)) / 4, ((-(f(s) - g(s))) * sqrt(3) / 4)), (s, f(s), 0), (s, f(s) - (f(s) - g(s)) / 4, ((f(s) - g(s)) * sqrt(3) / 4)), (s, g(s) + (f(s) - g(s)) / 4, ((f(s) - g(s)) * sqrt(3) / 4)), (s, g(s), 0)) → poly1, h_1, i_1, j_1, k_1, l_1, m_1
  2. Surface(u, (v * ((0.75 * g(u)) + (0.25 * f(u)))) + ((1 - v) * ((0.25 * g(u)) + (0.75 * f(u)))), ((f(u) - g(u)) * sqrt(3) / 4), u, -2, 2, v, 0, 1) → Top
  3. Surface(u, (v * ((0.25 * g(u)) + (0.75 * f(u)))) + ((1 - v) * f(u)), ((v * (f(u) - g(u))) * sqrt(3) / 4), u, -2, 2, v, 0, 1) → FrontTop
  4. Surface(u, (v * ((0.75 * g(u)) + (0.25 * f(u)))) + ((1 - v) * g(u)), ((v * (f(u) - g(u))) * sqrt(3) / 4), u, -2, 2, v, 0, 1) → BackTop
  5. Segment((s, f(s)), (s, g(s))) → n_1
  6. Sequence(Polygon[(-2 + (i * h), (3 * g(-2 + ((i + p) * h)) / 4) + f(-2 + ((i + p) * h)) / 4, ((g(-2 + ((i + p) * h)) - f(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + (i * h), (3 * f(-2 + ((i + p) * h)) / 4) + g(-2 + ((i + p) * h)) / 4, ((g(-2 + ((i + p) * h)) - f(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + (i * h), f(-2 + ((i + p) * h)), 0), (-2 + (i * h), (3 * f(-2 + ((i + p) * h)) / 4) + g(-2 + ((i + p) * h)) / 4, ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + (i * h), (3 * g(-2 + ((i + p) * h)) / 4) + f(-2 + ((i + p) * h)) / 4, ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + (i * h), g(-2 + ((i + p) * h)), 0)], i, 0, n - 1) → LeftHex
  7. Sequence(Polygon[(-2 + ((i + 1) * h), (3 * g(-2 + ((i + p) * h)) / 4) + f(-2 + ((i + p) * h)) / 4, ((g(-2 + ((i + p) * h)) - f(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), (3 * f(-2 + ((i + p) * h)) / 4) + g(-2 + ((i + p) * h)) / 4, ((g(-2 + ((i + p) * h)) - f(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), f(-2 + ((i + p) * h)), 0), (-2 + ((i + 1) * h), (3 * f(-2 + ((i + p) * h)) / 4) + g(-2 + ((i + p) * h)) / 4, ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), (3 * g(-2 + ((i + p) * h)) / 4) + f(-2 + ((i + p) * h)) / 4, ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), g(-2 + ((i + p) * h)), 0)], i, 0, n - 1) → RightHex
  8. Sequence(Polygon[(-2 + (i * h), (3 * f(-2 + ((i + p) * h)) / 4) + g(-2 + ((i + p) * h)) / 4, ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + (i * h), (3 * g(-2 + ((i + p) * h)) / 4) + f(-2 + ((i + p) * h)) / 4, ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), (3 * g(-2 + ((i + p) * h)) / 4) + f(-2 + ((i + p) * h)) / 4, ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), (3 * f(-2 + ((i + p) * h)) / 4) + g(-2 + ((i + p) * h)) / 4, ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 4))], i, 0, n - 1) → TopRects
  9. Sequence(Polygon[(-2 + (i * h), (3 * f(-2 + ((i + p) * h)) / 4) + g(-2 + ((i + p) * h)) / 4, ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), (3 * f(-2 + ((i + p) * h)) / 4) + g(-2 + ((i + p) * h)) / 4, ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), f(-2 + ((i + p) * h)), 0), (-2 + (i * h), f(-2 + ((i + p) * h)), 0)], i, 0, n - 1) → FrontTopRects
  10. Sequence(Polygon[(-2 + (i * h), (3 * f(-2 + ((i + p) * h)) / 4) + g(-2 + ((i + p) * h)) / 4, ((g(-2 + ((i + p) * h)) - f(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), (3 * f(-2 + ((i + p) * h)) / 4) + g(-2 + ((i + p) * h)) / 4, ((g(-2 + ((i + p) * h)) - f(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), f(-2 + ((i + p) * h)), 0), (-2 + (i * h), f(-2 + ((i + p) * h)), 0)], i, 0, n - 1) → FrontBotRects
  11. Sequence(Polygon[(-2 + (i * h), (3 * g(-2 + ((i + p) * h)) / 4) + f(-2 + ((i + p) * h)) / 4, ((g(-2 + ((i + p) * h)) - f(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), (3 * g(-2 + ((i + p) * h)) / 4) + f(-2 + ((i + p) * h)) / 4, ((g(-2 + ((i + p) * h)) - f(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), g(-2 + ((i + p) * h)), 0), (-2 + (i * h), g(-2 + ((i + p) * h)), 0)], i, 0, n - 1) → BackBotRects
  12. Sequence(Polygon[(-2 + (i * h), (3 * g(-2 + ((i + p) * h)) / 4) + f(-2 + ((i + p) * h)) / 4, ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), (3 * g(-2 + ((i + p) * h)) / 4) + f(-2 + ((i + p) * h)) / 4, ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), g(-2 + ((i + p) * h)), 0), (-2 + (i * h), g(-2 + ((i + p) * h)), 0)], i, 0, n - 1) → BackTopRects
  13. Sequence(Polygon[(-2 + (i * h), (3 * f(-2 + ((i + p) * h)) / 4) + g(-2 + ((i + p) * h)) / 4, ((g(-2 + ((i + p) * h)) - f(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + (i * h), (3 * g(-2 + ((i + p) * h)) / 4) + f(-2 + ((i + p) * h)) / 4, ((g(-2 + ((i + p) * h)) - f(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), (3 * g(-2 + ((i + p) * h)) / 4) + f(-2 + ((i + p) * h)) / 4, ((g(-2 + ((i + p) * h)) - f(-2 + ((i + p) * h))) * sqrt(3) / 4)), (-2 + ((i + 1) * h), (3 * f(-2 + ((i + p) * h)) / 4) + g(-2 + ((i + p) * h)) / 4, ((g(-2 + ((i + p) * h)) - f(-2 + ((i + p) * h))) * sqrt(3) / 4))], i, 0, n - 1) → BottomRects
  14. Integral(((3 * sqrt(3) / 8) * (f(x) - g(x))^(2)), -2, 2) → VolInt
  15. Surface(u, (v * ((0.75 * g(u)) + (0.25 * f(u)))) + ((1 - v) * g(u)), ((v * (g(u) - f(u))) * sqrt(3) / 4), u, -2, 2, v, 0, 1) → BackBot
  16. Surface(u, (v * ((0.75 * g(u)) + (0.25 * f(u)))) + ((1 - v) * ((0.25 * g(u)) + (0.75 * f(u)))), ((g(u) - f(u)) * sqrt(3) / 4), u, -2, 2, v, 0, 1) → Bottom
  17. Surface(u, (v * ((0.25 * g(u)) + (0.75 * f(u)))) + ((1 - v) * f(u)), ((v * (g(u) - f(u))) * sqrt(3) / 4), u, -2, 2, v, 0, 1) → FrontBot