Can anyone explain intuitively why increasing the circumference by 1 meter always increases the radius by 15.9cm?

I found the mathematical proof, and it is obviously correct. But how can the increase in radius be constant regardless of the starting circumference? With a very small circle, the increase should be huge but with a massive circle, the difference should end up miniscule shouldnt it? After all, the 1 m is getting distributed over a much larger circumference

Let the radius of the sphere be R and the new radius be R', hence

$2\pi R' = 2 \pi R + 1$

or,

$2\pi(R'-R) = 1$

or, the height $R' - R$ is $1/{2\pi} = 15.9 cms.$

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I think the physical reasoning will still be mathematical! –  Ali Jul 21 '13 at 15:34
Yeah, I'm not entirely sure if this is appropriate for Physics.SE. It seems to be asking for intuitive mathematical reasoning (!= physical reasoning) –  Manishearth Jul 21 '13 at 15:42
@Ali I was coming from the POV that this would be related to Newtonian Physics. But yeah, math does make more sense –  user87166 Jul 24 '13 at 14:09

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With a small circle, the increase is huge with respect to the small circle. Basically, for a circle of radius 1cm, the 15.9cm increase is a lot ... with respect to the original circle.

To convince yourself, consider this: When you increase the radius of a 1m radius circle by 1m, the radius doubles, but if you do it to a 100m circle, it just increases by a tiny amount relative to the original radius. So in the case of the larger circle, the increase seems much less because the relative increase is less, but in both cases the actual increase is the same.

It's a similar situation with the 15.9 cm increase. The absolute increase is the same, but the relative increase (which is what our brains find easier to imagine) is much smaller

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Relative vs absolute resolved 80% of my doubt. the remaining 20% is just me not being able to visualise how, adding a constant amount of circumference to any circumference increases the radius by a constant amount –  user87166 Jul 24 '13 at 14:11