A Perplexing Geometry Fun Fact

If a golf ball wanted to buy a belt, how big would it have to be? What if this golf ball was trying to gain a bit of weight to become a softball, and was planning on adding a few inches around the middle? How much longer should the belt be to ensure it will still fit?

What if it wasn’t a golf ball that wanted to buy a belt, but the entire Earth? How long would its belt be? And how much longer would its belt need to be if it too planned on gaining an extra inch or two around the middle?

The answers to these seemingly silly questions—and the relatively simple geometry at their heart—will surprise you. Read on to find out why!

The Circumference of a Circle

Before our golf ball can get all gussied up and wear his fancy new belt, the belt maker needs to figure out how big it should be. In other words, the belt maker needs to know how far it is all the way around the golf ball.

But if you think about it, you’ll see that this question isn’t specific enough. After all, I can imagine lots of different ways “around” a sphere like a golf ball, and each of these ways around has a different length.

To see what I mean, imagine making a bunch of different horizontal cuts through a golf ball. The outside edge of each of these cuts provides a different way to travel “around” the ball. Which do we want?

Well, you may not have known this, but when it comes to golf balls, you really want a belt to fit right around its middle—anything else just looks silly. Which means you really want to measure the distance around the ball’s equator (that’s the largest distance around that you can find) when you’re sizing it for a new belt.

The length of this equator is equal to the circumference of the circle whose radius is the same as the radius of the ball. So the length of the ball’s equator—and thus the length of the ball’s belt—is given by 2πr. For a standard sized golf ball with a radius of about 0.8 inches, that’s a circumference of about 5.03 inches.

From a Golf Ball to a Softball

If this cute little belt-wearing golf ball wanted to turn into a slightly more rotund belt-wearing softball, how much longer would it’s belt need to be? Well, a softball has a radius of about 1.8 inches (an inch more than a golf ball), which means the golf ball’s belt had better be at least as long as the circumference of a circle with a 1.8 inch radius, to ensure that it’s future-proofed against inflation.

The circumference of the circle with this new radius is given by 2π•1.8 inches, or about 11.31 inches. How much longer is this than the original distance around the golf ball? It’s 11.31 – 5.03 = 6.28 inches longer—more than double the size, just to add 1 inch all the way around!

The Circumference of the Earth

The length of the Earth’s looser, more comfortable belt is equal to the circumference of a circle with a radius that’s 1 inch longer than Earth’s 3,956 mile radius.

Let’s now think about getting the Earth sized up properly for its new belt. The Earth’s radius is (obviously) much bigger than the radius of a golf ball—it’s around 3,956 miles.

So how large does Earth’s new big boy belt need to be? Time to break out the circumference formula again, and find that Earth’s belt needs to be at least 2π•3,956 miles = 24,856 miles long. That’s a big belt!

Now, what if the Earth isn’t into a super tight-fitting belt like this, but instead would like a slightly comfier one with an extra inch of slack all the way around? In other words, how long would the belt need to be so that you could hold the it 1 inch off the ground all the way around the planet, at every point simultaneously? Surely that would require a much longer belt, don’t you think?

Maybe…or maybe not. Let’s think about it.

A New Belt for the Earth

The length of the Earth’s slightly looser and more comfortable belt is equal to the circumference of a circle with a radius that’s 1 inch longer than Earth’s 3,956 mile radius. So the amount of extra length in this new belt is this new circumference minus the Earth’s actual circumference—2π•(3,956 miles + 1 inch) – 2π•3,956 miles.

If you think about this expression, you’ll see that the actual circumference of the Earth—which is 2π•3,956 miles—amazingly subtracts away and has no impact on the length of the extra bit of belt. This says that the entire belt will have 1 inch of slack all the way around the planet if we add only 2π•1 inch, or about 6.28 inches, to its total length!

As we’ve seen, the geometry says that the radius of the ball we’re wrapping a belt around simply doesn’t matter. No matter how big or small it is, we always need to add just 6.28 inches more in order for it to have 1 inch of slack all the way around.

For a golf ball, that 6.28 inches of extra belt is a lot. But for something the size of the Earth, it’s a startlingly small amount. Pretty perplexing, right?

Wrap Up

Okay, that’s all the geometry fun we have time for today.

For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

Earth image from Shutterstock.