The Circle Game

Ok, so this is really not a riddle or puzzle. It's just a fact. In fact, one of the facts of nature I find the absolutely most un-intuititive.

So here is the question: Imagine if you tied a string around the world and stretched it tight till you circled the Earth once, so it was tight against the surface of the planet. Then take that same string and added 3 feet to its length. If you stretched it around the Earth again, there would be some slack it in this time. If you were to space that slack evenly, how far off the surface of the planet would the string be?

The answer is 6 inches. Yes, adding 3 feet of string to the string actually makes the string raise 6" off the surface of the planet all around the entire planet. Hard to believe huh?

I think so. Well here's the proof.

Proof:
The string without the 3 feet added represents the circumference of the Earth, let's call it X. We are now changing! X to be X + 3. If the string was initially stretched tight to the surface, then its distance from the center of the planet is basically the radius. Let's call it R. The question is really how does the radius from the center of the planet change if the circumference is changed from X to X + 3?

Doing the geometry, initially the radius was r and the circumference of the planet was 2 * Pi * r. Once we add 3 feet to the circumference, it becomes = (2 * Pi * r) + 3. To generically solve for radius, we divide circumference by 2Pi. Dividing the new circumference of (2 * Pi * r) + 3 by 2Pi will give us the new radius of r'. Solving this equation gives us:
(2 Pi r)/(2Pi) + 3/(2Pi) = r + 3/(2Pi) where r = the original radius, or roughly r + 3/6. This means the new radius after adding 3 feet to the circumference results in a new radius from the center of the circle of the original radius plus 3/6 or 1/2 of a foot or 6 inches.

Yes, it is hard to believe, but ! adding 3 feet to the circumference of ANY circle results in a ! radius 6 " large. Even if you wrap a string a 3 foot string around a pin point, adding 3 more feet to the string will increase the radius by 6". The same is true if you due it to the circumference of a basketball or the planet Earth. How can this be?

It seems so unintuitive, though the truth is in the proof above. One way to think about it to make it a bit more intuititve is that while 6" is always the absolute change in the length of the radius regardless of the size of the object, the 6" gain in radius length is a different percentage change in the radius depending on the object size. To a 3 feet total string length (eg. a basketball), adding 3 feet to the string will add 6" inches to the radius - which almost doubles the length of the radius. Adding 3 feet to the circumference of planet Earth (24901 miles) also increases the radius by 6", but relative to the original length of the radius (3963 miles) this is a miniscual relative increase.

Circumference of the circle