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Hemisphere Volume Calculation

Hemisphere Volume Calculation
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2016-03-02 16:01
Solid GeometryCalculusGeometric Construction
This example demonstrates how to construct a hemisphere and calculate its volume using GeoGebra. Key math concepts include solid geometry, volume calculation formulas, and 3D shape construction. GeoGebra commands/objects used in the construction include Circle, Point, Sphere, Cylinder, Cone, Segment, OrthogonalPlane, IntersectPath, Intersect, Rotate, Polygon, Sequence, Angle, and Surface. It is suitable for high school students learning solid geometry, helping them understand the derivation of the hemisphere volume formula and the construction of 3D shapes.

GeoGebra Commands & Parameters

  1. Circle(A, 1) → c
  2. Circle(B, 1) → d
  3. Point(c) → C
  4. Sphere(A, C) → a
  5. Cylinder(B, D, 1) → b, e, f, g
  6. Cone(D, B, 1) → h, k, i
  7. Segment(E, E + (0, 0, 1)) → j
  8. Point(j) → F
  9. OrthogonalPlane(F, j) → l
  10. IntersectPath(l, a) → p
  11. IntersectPath(l, h) → q
  12. IntersectPath(l, b) → r
  13. Point(q) → H
  14. Intersect(Ray[G, H], r) → I,
  15. Rotate(H, α, G) → H'
  16. Rotate(I, α, G) → I'
  17. Polygon(H, H', I', I) → poly1, h_1, h', i', i_1
  18. Sequence(Rotate[poly1, (k * α), G], k, 0, n) → list1
  19. Intersect(Ray[J, J + (1, 0, 0)], p) → K,
  20. Segment(A, J) → s
  21. Segment(J, K) → t
  22. Segment(K, A) → a_1
  23. Intersect(Ray[G, G + (1, 0, 0)], q) → L, M
  24. Segment(B, G) → b_1
  25. Segment(G, L) → c_1
  26. Circle(s, C) → d_1
  27. Angle(A, J, K) → β
  28. Angle(B, G, L) → γ
  29. Circle(B, 1, Segment[B, B + (0, 0, 1)]) → e_1
  30. Surface((sqrt(1 - h^(2)) * cos(θ)) + x(A), (sqrt(1 - h^(2)) * sin(θ)) + y(A), h, θ, 0, 6.28319, h, 0, 1) → o
  31. Point(e) → N
  32. Segment(B, N) → m