If you'd ask the Ordnance Survey (the mapping authority for the United Kingdom), they might give you a number of 11,073 miles (17,820 km). That's all well and good, but what does this number actually mean?
If I said the coastline had an infinite length, would I be wrong?
To illustrate the problem imagine a huge printed map of Britain, billboard size. On it, you could measure the outline of Britain with a desk ruler (30cm). A quick approximation can be made by moving the whole thing around the perimeter. This way you measure distances of 30cm at a time; sum it all up and you have a rough figure. If you want more precision, use the ruler to measure 1cm at a time instead. This smaller step-size is a lot more work but it would result in a larger end result for sure.
So which figure is correct? I might say more precision is better, but then what is the limit? You could get even more precision by moving down to millimeters. Then nanometers. Then atomic or even subatomic scales. Eventually, any real world possibility of measuring is gone. But you lost any usable number long before that.
Mathematically there is no such physical limit. You could increase the precision forever, and the measured length of the coastline would increase along with it. Like a snowflake that offers more detail every time you look closer.
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