Calculate the height of the tortoise

I said 2pi (in what ever unit you are using) as the extra length required in the rope for the rope to be uniformly 1 unit above the circumference of the circular object, in this case the Earth.
So for the rope to be 1m above the earth all the way around (assuming it to be perfectly circular) it only needs to be 6.28… metres longer that one that is a snug fit.

Astonishing when the Earth has a circumference of about 40 000 000 metres.

This number holds true for circles of all sizes.
Based on the information in the original question (where the diameter of the earth was provided), I would say the answer is one pi.
Circumference of circle is pi D (obviously equivalent to 2pi(r).
 
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