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

[rbwang1225]

Homework Statement

By employing spherical polar coordinates show that the circumference C of a circle of radius R inscribed on a sphere $S^{2}$ obeys the inequality C<2$\pi$R

The Attempt at a Solution

I proved C=2$\pi$R$\sqrt{1-\frac{R^2}{4r^2}}$

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.

 

[Simon Bridge]

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).