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Hexagon Cross Section Volume Calculation

Hexagon Cross Section Volume Calculation
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2018-03-19 02:00
PolygonFunctionSolid GeometryCalculusIntegral
In collections intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve Volumes in Calculus Volumes in Calculus
This example demonstrates how to calculate the volume using a hexagon cross-section diagonal. It involves core math concepts like volume, step functions, exponentiation, arithmetic mean, scatter plots, and roots. Construction highlights include commands like Polygon, Surface, Segment, Sequence, and Integral in GeoGebra. Suitable for high school students and math enthusiasts studying solid geometry and calculus.

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

AI Drawing Prompt Reference

Create a model demonstrating volume calculation using a hexagon cross-section diagonal. The model features multiple surfaces and rectangles that form the upper and lower parts of the hexagon, along with front and back sections. Start by drawing the hexagon cross-section, then construct the upper and lower surfaces. Add rectangles for the front and back surfaces, and finally, calculate the volume through integration. The model should illustrate key math concepts like volume, step functions, exponentiation, arithmetic mean, scatter plots, and roots.
The prompt above was compiled from the original drawing commands and is for reference only.