# Solution to Riddle of the Week: The Long Belt Problem

Did you solve it?

Kory Kennedy using Illustration Copyright csaimages.com

This is a solution to To Solve This Twisty Math Riddle, You Just Need One Belt and One Earth, part of our Riddle of the Week series.

### To Solve This Twisty Math Riddle, You Just Need One Belt and One Earth

If you add a mere 6 feet to a belt that’s 130 million feet long, you’re adding a minuscule 0.000005 percent to its length. Accordingly, my intuition says you wouldn’t even be able to notice the difference between the two belts when they’re wrapped around the earth. But my intuition is completely wrong. Perhaps yours is better!

Let’s call Earth’s radius “R,” measured in feet. (We don’t actually need to know the radius is about 4,000 miles, or 20 million feet.) This means your belt is exactly the length of Earth’s circumference:

Peter’s belt is 6 feet longer than yours:

Peter’s belt = 2*𝛑*R+6

Imagine pulling up on Peter’s belt so it’s truly suspended above Earth’s surface. The question at hand is how high above Earth’s surface is it? Or, in other words, how much longer is the radius of the circle that it encompasses?

Let’s call the height we’re looking for “H.” Then, the radius of the circle that Peter’s belt encompasses is R+H. So the length of Peter’s belt must be its circumference, or: 2*𝛑*(R+H). But we already know Peter’s belt is 2*𝛑*R+6.

Eureka! We have an equation!

2*𝛑*R+6 = 2*𝛑*(R+H)

Amazingly, the 2*𝛑*R drops out on both sides, and we get:

H = 3/𝛑

Since 𝛑 is about 3, H is about 1 foot! Or, to be exact, H is 0.95 feet. Incredible!