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Volumes with Known Cross Sections

Volumes with Known Cross Sections
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2019-04-06 04:17
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates how to calculate the volume of a solid using known cross sections. Key mathematical concepts include solid geometry and similar triangles. Using GeoGebra commands such as Plane, Segment, Intersect, etc., the construction of the solid and dynamic demonstration are achieved. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Plane((0, 0, 0), (0, 1, 0), (0, 1, 1)) → s
  2. Plane((a, 0, 0), (a, 1, 1), (a, 1, 2)) → c
  3. Plane((b, 0, 0), (b, 1, 1), (b, 1, 2)) → d
  4. Segment((a, f(a)), (a, g(a))) → h
  5. Segment((b, f(b)), (b, g(b))) → i
  6. Intersect(f, h) → A
  7. Intersect(g, h, (-8, -6.968074739870145)) → B
  8. Intersect(f, i) → C
  9. Intersect(g, i, (8, -1.0319252601298547)) → D
  10. Polygon(A, B, (x(B), y(B), h), (x(A), y(A), h)) → q1, a_1, b_1, t, f_1
  11. Polygon(A, B, (a, (0.5 * (y(A) + y(B))), (sqrt(3) * h / 2))) → t1, g_1, a_2, b_2
  12. Midpoint(A, B) → E
  13. Sphere(E, A) → u
  14. Sequence(Polygon[(a + (i * deltax), g(a + (i * deltax)), 0), (a + (i * deltax), f(a + (i * deltax)), 0), 4, s], i, 0, n - 1, 1) → L3
  15. Polygon(A, B, (x(B), y(B), h)) → t2, k, a_3, b_3
  16. Sequence(Polygon[(a + (i * deltax), g(a + (i * deltax)), 0), (a + (i * deltax), f(a + (i * deltax)), 0), 3, s], i, 0, n - 1, 1) → L4
  17. Sequence(Polygon[(a + (i * deltax), f(a + (i * deltax)), 0), (a + (i * deltax), g(a + (i * deltax)), 0), (a + (i * deltax), g(a + (i * deltax)), Distance[(a + (i * deltax), f(a + (i * deltax)), 0), (a + (i * deltax), g(a + (i * deltax)), 0)])], i, 0, n - 1, 1) → L7
  18. Line((0, 0, 0), (1, 0, 0)) → m
  19. Line((0, 0, 0), (0, 1, 0)) → r
  20. Textfield(f) → InputBox1
  21. Textfield(g) → InputBox2
  22. Textfield(a) → InputBox3
  23. Textfield(b) → InputBox4
  24. Textfield(n) → InputBox5
  25. Sequence(Prism[Element[L3, i], deltax], i, 1, n, 1) → L2
  26. Sequence(Prism[Element[L4, i], deltax], i, 1, n, 1) → L5
  27. Sequence(Polygon[(a + (i * deltax), g(a + (i * deltax)), 0), (a + (i * deltax), f(a + (i * deltax)), 0), 5, s], i, 0, n - 1, 1) → L6
  28. Sequence(Prism[Element[L6, i], deltax], i, 1, n, 1) → L8
  29. Sequence(Midpoint[(a + (i * deltax), f(a + (i * deltax)), 0), (a + (i * deltax), g(a + (i * deltax)), 0)], i, 0, n - 1, 1) → L9
  30. Sequence((a + (i * deltax), f(a + (i * deltax)), 0), i, 0, n - 1, 1) → L10
  31. Sequence(Segment[Element[L9, i], Element[L10, i]], i, 1, n, 1) → radii
  32. Sequence((a + (i * deltax), g(a + (i * deltax)), 0), i, 0, n - 1, 1) → L11
  33. Sequence(Midpoint[Element[L10, i], Element[L11, i]], i, 1, n, 1) → L12
  34. Sequence(CircleArc[Element[L12, i], Element[L10, i], Element[L11, i], s], i, 1, n, 1) → L13
  35. Sequence(Prism[Element[L7, i], (-deltax)], i, 1, n, 1) → L