A classic problem about circles that defies intuition: the result does not depend on what one would expect.
The rope around the Earth
Riddle statement
A rope surrounds the Earth adjusted to the equator. Then it is lengthened exactly 2\pi meters and placed again forming a concentric circle, uniformly spaced from the ground.
How much does the rope rise above the surface?
Show solution
Solution
Answer: it rises 1 meter.
Let $R$ be the radius of the Earth. The initial chord measures:
$
2\pi R.
$
When uniformly rising to a height $h$, the new radius is:
$
R+h.
$
The new length is:
$
2\pi(R+h).
$
As the rope has lengthened $2\pi$ meters:
$
2\pi(R+h)-2\pi R=2\pi.
$
By simplifying:
$
2\pi h=2\pi.
$
By both:
$
h=1.
$
So the rope is 1 meter from the ground throughout the entire loop.