A cord around the equator

For christmas I got a little book called math for the vest pocket (in german: Mathematik für die Westentasche) by the math Professor Albrecht Beutelspacher.
It is a popular science book and consists of 52 short topics on general math from "golden section" over "Fermat's last Theorem" to "Traveling Salesman Problem". Today I read the topic about "a cord around the equator".
The Author asks, if you would extend a cord around the equator by one meter (1m), what would be the difference between the radius of the original cord (equator) and the extended cord, which is also a circle with the same center point (earth center)?

He argues this way:
R-r = (ec+1)/(2*Pi)-ec/(2*Pi)
= ((ec+1)-ec)/(2*Pi)
= 1/(2*Pi)
= 0,159...

With:
ec = equator circumference (2 * Pi * r)
r = radius of the smaller circle (earth radius)
R = radius of the bigger circle (ec + 1 = 2 * Pi * R)

So by this the radius would extend by ~16 cm!!!
This is very stunningly to me and I could not really believe it.
So I sat down and computed it on a piece of paper. I did the following:
I assumed the earth radius is 6371000 m.
By this the circumference is 2*Pi*6371000 = 40030139,78 m.
Now I extended the Radius by 15,9... cm.
And we get exactly 2*Pi*6371000,159.. = 40030140,78!! One meter (1m) difference.

I still have problems to belief that... in other words, if you extend the circumference of a circle by one meter, the radius will extend by ~16cm, irrelevant how big the radius/circumference is. Because radius/circumference is 2*Pi... :D (damn still can't belief it!!!)
I really think about doing an experiment and measure this stuff out... in a smaller scale for sure. ;-)