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Napkin Ring Problem

Napkin Ring Problem
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2020-06-22 04:46
CircleSolid GeometryParametric EquationGeometric Construction
This example demonstrates the Napkin Ring Problem, illustrating that the surface area of a torus formed by rotating a ring is equal to that of a sphere with the same radius. It involves core mathematical concepts such as cylinder, sphere, intersection, fractions, circumcircle, difference and slope, and rotation. The construction uses commands like Sphere, Line, Segment, Point, Mirror, IntersectPath, Intersect, CircleArc, Surface, Distance, Cylinder, Midpoint, PointIn, and Vector, highlighting geometric construction techniques. Suitable for high school solid geometry education, it helps students understand rotational solids and surface area calculations.

GeoGebra Commands & Parameters

  1. Sphere(A, R1) → a
  2. Sphere(B, R2) → b
  3. Line(A, zAxis) → f
  4. Line(B, zAxis) → g
  5. Segment(C, A) → h1
  6. Segment(A, D) → i
  7. Segment(E, B) → j
  8. Segment(B, F) → k
  9. Point(h1) → G
  10. Mirror(G, A) → H
  11. IntersectPath(p, a) → c
  12. IntersectPath(q, a) → d
  13. IntersectPath(p, b) → e
  14. IntersectPath(q, b) → r
  15. Line(G, xAxis) → l
  16. Line(H, xAxis) → m
  17. Intersect(l, c) → I, J
  18. Intersect(m, d) → K, L
  19. CircleArc(A, K, I) → s
  20. Intersect(l, b) → M, N
  21. Intersect(m, r) → O, P
  22. CircleArc(B, O, N) → t
  23. Surface(s, (2 * π), f) → n
  24. Surface(t, (2 * π), g) → o
  25. Distance(G, H) → h
  26. Cylinder(d, h) → u, c_1, v
  27. Cylinder(r, h) → w, d_1, a_1
  28. Midpoint(M, N) → Q
  29. Midpoint(G, Q) → R
  30. Midpoint(P, O) → S
  31. Midpoint(H, S) → T
  32. Midpoint(R, T) → U
  33. Segment(I, M) → f_1
  34. PointIn(v) → V
  35. Segment(V, P) → g_1
  36. Vector(U, R) → u_1
  37. Vector(U, T) → v_1