Band around the Earth Puzzle - Solution

The Puzzle:

Band around the Earth
The circumference of the Earth is approximately 40,000 kilometers, and someone has just made a metal band that circles the Earth, touching the ground at all locations.

You come along at night, as a practical joke, and add just 10 meters to its length (one hundredth of one kilometer !)

It is now one four-millionth longer, and sits magically just above the ground at all locations

How far has it risen ... could a flea, a rabbit or even a man squeeze underneath it?

Our Solution:

Use the formula Circumference = 2 × pi × Radius

Before: Original Circumference = 2 × pi × R
After: Original Circumference + 10m = 2 × pi × (R + Gap)

Subtracting the two:

10m = 2 × pi × Gap

So, the Gap = 10m / (2 × pi) = 1.6m approximately

So a man could fit under it easily (though he might bump his head)

Note: Adding 10m to the circumference of ANY circle increases the radius by 10m / (2 × pi), no matter what the original circumference was.

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