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Volume of a Torus

Volume of a Torus
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2019-03-28 00:51
Solid GeometryCalculusIntegral
This example demonstrates the volume calculation of a solid of revolution using the washer method. Key mathematical concepts include the washer volume formula and the geometric construction of solids of revolution. Using GeoGebra commands such as Invert, Rotate, and Surface, the process of calculating the volume of a torus is dynamically illustrated. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Invert(g1) → i1
  2. Invert(g2) → i2
  3. Line(A, B) → L
  4. Rotate(f1, θ, L) → p1
  5. Line(C, D) → M
  6. Rotate(f1, θ, M) → q1
  7. Textfield(c) → inputBox1
  8. Textfield(d) → inputBox2
  9. Textfield(g1) → inputBox3
  10. Textfield(b) → inputBox5
  11. Textfield(a) → inputBox6
  12. Rotate(f2, θ, L) → p2
  13. Rotate(f2, θ, M) → q2
  14. Textfield(g2) → inputBox4
  15. Surface(u, ((f1(u) - c) * cos(w)) + c, ((f1(u) - c) * sin(w)), u, a, b, w, 0, θ) → pf1
  16. Surface(u, ((f2(u) - c) * cos(w)) + c, ((f2(u) - c) * sin(w)), u, a, b, w, 0, θ) → pf2
  17. Segment(E, F) → EF
  18. Segment(G, H) → GH
  19. Rotate(E, θ, L) → E'
  20. Rotate(F, θ, L) → F'
  21. Segment(E', F') → EF'
  22. Rotate(G, θ, L) → G'
  23. Rotate(H, θ, L) → H'
  24. Segment(G', H') → GH'
  25. Surface(x(F), ((y(F) - c) * cos(t)) + c + ((s * (f1(x(E)) - f2(x(F)))) * cos(t)), ((y(F) - c) * sin(t)) + ((s * (f1(x(E)) - f2(x(F)))) * sin(t)), s, 0, 1, t, 0, θ) → lcap
  26. Surface(x(H), ((y(H) - c) * cos(t)) + c + ((s * (f1(x(G)) - f2(x(H)))) * cos(t)), ((y(H) - c) * sin(t)) + ((s * (f1(x(G)) - f2(x(H)))) * sin(t)), s, 0, 1, t, 0, θ) → rcap