Imagine the Earth was perfectly smooth and round, and at the Equator there was a metal band that circumscribed it like a metal hoop around the middle of a wooden barrel. Although the metal band is 24,901 miles long—that’s the circumference of the Earth at the Equator—it’s a perfect fit, so snug you can’t even slip a playing card under it. Now here’s the question: If you wanted to raise the metal band two feet so that a person could slip under it from one hemisphere to the other, how much would you have to increase the overall circumference of the metal band?

Thank you Taylor for this interesting math riddle submission!

My father taught me this, but I never believed the answer. Then I worked it out mathematically, and it makes sense. But it’s still unbelievable.

1 mile = 5280 feet

24,901 miles = 131477280 feet, is circumference of band

Diameter of band = 41850518.03255438 feet

We want to increase the diameter by 4 feet, 2 feet on each side

New diameter = 41850522.03255438

New circumference = 131477292.56637062 feet

Should double check math, but logic is sound

2pi

I meant 4pi

Taylor, do you think Frank’s answer 2pi is correct?

Basically, yes! A little over 6 feet is all you would need to add to the circumference of the world in order to increase the diameter of the world by two feet (2 feet * pi). If you wanted to increase the diameter four feet (two feet all the way around the world, which is not what the puzzle asks for) it would be twice that, or about 12.5 feet.

Frank, you are today’s winner!

2•Pi

2 pi makes no sense. How does the unit of of length get identified? 2pi inches? Feet? Miles? Is the suggestion that the answer is the same for a circle of any size? I never stated the difference between the old and new circumference in my response, and didn’t double check math, but arrived at just over 12.5 feet. Simple geometry. You want to increase the diameter by 4 feet so what do you need to add to the circumference? 12.56637062 feet.

Wait, Taylor says increase the diameter by 2 feet, that would be just over 6 feet to circumference. Coincidence that 2pi feet is close to answer. However the riddle says to have the band 2 feet off the ground. That means the diameter must be increased by 4 feet because it’s two feet off the ground on both sides of the planet.

Crabman, Frank’s answer is in feet, and he should have said so. At first, I was like you, and I crunched this hard on my calculator using about 15 digits of pi. But the answer I came up with was 6.283 feet, which is exactly pi times two. After reviewing my calculations and moving a few things around using the commutative property, I realized it was no coincidence! If you want to increase the diameter of the earth by 2 feet, you need to increase the circumference of the earth by 2 feet times pi. If you want to increase the diameter by 4 feet so that the new circumference is uniformly 2 feet above the earth all around the world (which is not what the puzzle asks for nor is what is pictured in the graphic) The answer is four times pi feet. Does that make sense?

Circumference = pi x d, so yes it does make perfect sense that to increase d by 2 feet you would increase the circumference by 2feet x Pi which is exactly the math I posted (except I incorrectly interpreted 2 feet off the ground in all locations. Not sure why I thought/posted it was coincidence.

Congrats Frank for quickly and correctly arriving at the simple answer.