Band Around the World Puzzle

To solve this, you can use this formula: $\mathrm{circumference}=2\times \pi \times \mathrm{radius}$

Or for short: $c=2\pi r$

This can be changed to

$$r=\frac{c}{2\pi }$$

The original band circumference was the same as that of the Earth, 40,000,000 meters.

Using the above formula we can calculate the original band radius. Let’s solve to 1 decimal place.

$$r=\frac{\mathrm{40,000,000} m}{2\pi }$$ $$r=\mathrm{6,366,197.7} m$$

The radius of the band is approximately 6,366,197.7 meters.

When 10 meters of band is added, the new circumference will be 40,000,010 meters.

Let’s find the new radius.

$$r=\frac{\mathrm{40,000,010} m}{2\pi }$$ $$r=\mathrm{6,366,199.3} m$$

The difference between the new radius and the old radius shows how high the band will be above the ground after the new length is added.

$$\mathrm{6,366,199.3} m-\mathrm{6,366,197.7} m=1.6 m$$

After adding 10 meters to the band's length, it will be raised 1.6 meters off the ground!

This answer is somewhat counterintuitive because, for many people, the intuitive answer is that it won’t rise a noticeable amount off the ground.

What’s even more surprising is that the answer is the same no matter the size of the sphere!

This means you didn’t actually need to know the circumference of the Earth to solve this puzzle. You can calculate the change in radius of a circle given a change in circumference using this formula:

$$\mathrm{\Delta r}=\frac{\mathrm{\Delta c}}{2\pi }$$