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Hexagon Volume Calculation Demo

Hexagon Volume Calculation Demo
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2018-03-20 23:19
PolygonSolid GeometryCalculusIntegral
In collections Volumes in Calculus
This example demonstrates the calculation of volume based on a hexagonal base, involving core math concepts like integral calculus, solid geometry, and polygon area. It utilizes GeoGebra commands and objects such as Integral, Surface, Sequence, and Polygon as per the construction summary, making it suitable for high school and above students learning the integration of solid geometry and calculus.

GeoGebra Commands & Parameters

  1. Integral(((3 * sqrt(3)) * (f(x) - g(x))^(2) / 2), -2, 2) → Vol
  2. Surface(u, f(u) + (v * (f(u) - g(u)) / 2), ((v * (f(u) - g(u))) * sqrt(3) / 2), u, -2, 2, v, 0, 1) → FrontBottom
  3. Surface(u, (v * g(u)) + ((1 - v) * f(u)), ((f(u) - g(u)) * sqrt(3)), u, -2, 2, v, 0, 1) → Top
  4. Surface(u, (v * g(u)) + ((1 - v) * f(u)), 0, u, -2, 2, v, 0, 1) → Base
  5. Surface(u, f(u) + (f(u) - g(u)) / 2 - (v * (f(u) - g(u)) / 2), (((1 - v) * (f(u) - g(u))) * sqrt(3) / 2) + ((v * (f(u) - g(u))) * sqrt(3)), u, -2, 2, v, 0, 1) → FrontMiddle
  6. Sequence(Polygon[(-2 + (i * h), g(-2 + ((i + p) * h)), 0), (-2 + (i * h), f(-2 + ((i + p) * h)), 0), (-2 + (i * h), (1.5 * f(-2 + ((i + p) * h))) - (0.5 * g(-2 + ((i + p) * h))), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 2)), (-2 + (i * h), f(-2 + ((i + p) * h)), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3))), (-2 + (i * h), g(-2 + ((i + p) * h)), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3))), (-2 + (i * h), (1.5 * g(-2 + ((i + p) * h))) - (0.5 * f(-2 + ((i + p) * h))), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 2))], i, 0, n - 1) → LeftHex
  7. Sequence(Polygon[(-2 + ((i + 1) * h), g(-2 + ((i + p) * h)), 0), (-2 + ((i + 1) * h), f(-2 + ((i + p) * h)), 0), (-2 + ((i + 1) * h), (1.5 * f(-2 + ((i + p) * h))) - (0.5 * g(-2 + ((i + p) * h))), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 2)), (-2 + ((i + 1) * h), f(-2 + ((i + p) * h)), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3))), (-2 + ((i + 1) * h), g(-2 + ((i + p) * h)), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3))), (-2 + ((i + 1) * h), (1.5 * g(-2 + ((i + p) * h))) - (0.5 * f(-2 + ((i + p) * h))), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 2))], i, 0, n - 1) → RightHex
  8. Sequence(Polygon[(-2 + (i * h), g(-2 + ((i + p) * h)), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3))), (-2 + (i * h), f(-2 + ((i + p) * h)), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3))), (-2 + ((i + 1) * h), f(-2 + ((i + p) * h)), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3))), (-2 + ((i + 1) * h), g(-2 + ((i + p) * h)), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3)))], i, 0, n - 1) → TopRects
  9. Sequence(Polygon[(-2 + (i * h), f(-2 + ((i + p) * h)), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3))), (-2 + (i * h), (1.5 * f(-2 + ((i + p) * h))) - (0.5 * g(-2 + ((i + p) * h))), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 2)), (-2 + ((i + 1) * h), (1.5 * f(-2 + ((i + p) * h))) - (0.5 * g(-2 + ((i + p) * h))), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 2)), (-2 + ((i + 1) * h), f(-2 + ((i + p) * h)), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3)))], i, 0, n - 1) → FrontTopRects
  10. Sequence(Polygon[(-2 + (i * h), (1.5 * f(-2 + ((i + p) * h))) - (0.5 * g(-2 + ((i + p) * h))), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 2)), (-2 + (i * h), f(-2 + ((i + p) * h)), 0), (-2 + ((i + 1) * h), f(-2 + ((i + p) * h)), 0), (-2 + ((i + 1) * h), (1.5 * f(-2 + ((i + p) * h))) - (0.5 * g(-2 + ((i + p) * h))), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 2))], i, 0, n - 1) → FrontBotRects
  11. Sequence(Polygon[(-2 + (i * h), f(-2 + ((i + p) * h)), 0), (-2 + (i * h), g(-2 + ((i + p) * h)), 0), (-2 + ((i + 1) * h), g(-2 + ((i + p) * h)), 0), (-2 + ((i + 1) * h), f(-2 + ((i + p) * h)), 0)], i, 0, n - 1) → BottomRects
  12. Sequence(Polygon[(-2 + (i * h), g(-2 + ((i + p) * h)), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3))), (-2 + (i * h), (1.5 * g(-2 + ((i + p) * h))) - (0.5 * f(-2 + ((i + p) * h))), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 2)), (-2 + ((i + 1) * h), (1.5 * g(-2 + ((i + p) * h))) - (0.5 * f(-2 + ((i + p) * h))), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 2)), (-2 + ((i + 1) * h), g(-2 + ((i + p) * h)), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3)))], i, 0, n - 1) → BackTopRects
  13. Sequence(Polygon[(-2 + (i * h), g(-2 + ((i + p) * h)), 0), (-2 + (i * h), (1.5 * g(-2 + ((i + p) * h))) - (0.5 * f(-2 + ((i + p) * h))), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 2)), (-2 + ((i + 1) * h), (1.5 * g(-2 + ((i + p) * h))) - (0.5 * f(-2 + ((i + p) * h))), ((f(-2 + ((i + p) * h)) - g(-2 + ((i + p) * h))) * sqrt(3) / 2)), (-2 + ((i + 1) * h), g(-2 + ((i + p) * h)), 0)], i, 0, n - 1) → BackBotRects
  14. Surface(u, g(u) + (v * (g(u) - f(u)) / 2), ((v * (f(u) - g(u))) * sqrt(3) / 2), u, -2, 2, v, 0, 1) → BackBottom
  15. Surface(u, g(u) + (g(u) - f(u)) / 2 - (v * (g(u) - f(u)) / 2), (((1 - v) * (f(u) - g(u))) * sqrt(3) / 2) + ((v * (f(u) - g(u))) * sqrt(3)), u, -2, 2, v, 0, 1) → BackMiddle
  16. Polygon((s, f(s), 0), (s, (1.5 * f(s)) - (0.5 * g(s)), ((f(s) - g(s)) * sqrt(3) / 2)), (s, f(s), ((f(s) - g(s)) * sqrt(3))), (s, g(s), ((f(s) - g(s)) * sqrt(3))), (s, (1.5 * g(s)) - (0.5 * f(s)), ((f(s) - g(s)) * sqrt(3) / 2)), (s, g(s), 0)) → Slice, i, j, k, l, q, r