Suppose you tie a rope tightly around the Earth at the equator. (Assume the Earth is perfectly spherical, and that the surface is smooth so that the rope lies tight against the surface at all points.) Now suppose that you add an additional 6 feet to the length of the rope. How high off the surface would the rope lie? You could look up the Earth's circumference and do the math to come up with an exact answer, but can you quickly come up with an intuitive guess? (High enough to slide a piece of paper under? To wave your hand under? To walk under?) Click below to see a hint or the answer.
Hint: If I reversed the parameters and told you that I increased the length enough to raise the rope 6 feet from the surface in all directions, could you tell me how much was added to the length of the rope? (Given the formula for the circumference of a circle, C = 2πr, but not knowing the circumference of the Earth, can you come up with a guess?) Reversing that, can you come up with the answer to the original problem?
Answer: The first time they hear this puzzle, many people will try to do the math starting with the circumference of the Earth. That doesn't matter though. It's a property of any circle that if you increase the circumference by a fixed amount, the radius will change by that amount divided by 2π (because r = C/2π). The rope could be tied around a beach ball or a tennis ball and the answer would not change. So the exact answer to the problem is 0.95493 feet, but if you said "about 1 foot" you were right.