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Circumference C of a circle of radius R inscribed on a sphere

  1. Jan 22, 2012 #1
    1. The problem statement, all variables and given/known data
    By employing spherical polar coordinates show that the circumference C of a circle of radius R inscribed on a sphere [itex]S^{2}[/itex] obeys the inequality C<2[itex]\pi[/itex]R

    3. The attempt at a solution

    I proved C=2[itex]\pi[/itex]R[itex]\sqrt{1-\frac{R^2}{4r^2}}[/itex]

    So if r>R, then the equality is correct.

    Am I right? Since the statement of the problem doesn't give me the radius r of the sphere, I doubt my result.
     
  2. jcsd
  3. Jan 22, 2012 #2

    Simon Bridge

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    you mean something like:
    use spherical-polar and put the z axis through the center of the circle.
    the circle will be a line of constant θ from the z-axis.
    for a sphere radius R, the radius of the circle is r = Rθ, but the circumference is C=2πR.sinθ < 2πr.

    eg - biggest circle is a grand circle, r=R, θ=π/2, so C=2πR < 2πr=ππR
    the only time you get close is for θ → 0 (small circle).
     
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