Start out with the following simple question:
Imagine you have tied a rope tightly around the full span of earth's equator (assume "earth" as being a perfect sphere and that the rope cannot be stretched). Now imagine that you add 1 meter to the initial rope length so that it becomes loose around the equator. Question: if you spread the rope at an even height from earth's surface (i.e. rope has a circular form, cocentric with earth's center) can you pass a cat underneath it?
Intuition says "no way", but let's examine the problem closely. Cutting a 2D slice at the equator, initially we have a circle with a perimeter equal to earth's perimeter at the equator (call it's radius "R") and a circle of rope tied tightly around it. When we loosen the perimeter by a meter (add a meter of rope) we are (indirectly) increasing it's radius. What we need to find out is how much the circle's radius will expand. (again intuition says it'll be very little due to the earth's dimensions and the fact that we are adding so little rope).
So, initially we have, from basic geometry:
(where P is the initial perimeter). When we add a meter of rope we are making P'=P+1 (where P' is the perimeter after the meter of rope was added). We also know (same as equation (1)) that:
Solving for R':
note the beauty of equation (2). It tells us that the increase in the radius is independant of it's initial radius. Simply, it doesn't matter whether we are tying a rope around an orange or the earth, when we add 1 meter to it's length (perimeter) , we ALWAYS increase it's radius by 16 centimeters.
let's just say I was marvelled by this simple result.
by the way... most cats can squeeze beneath 16 centimeters.