# Math behind “Rope around the globe”

Some people will know the mathematical problem:

Take a rope and tie it around a tennis ball very tightly. Cut the rope and add an extra meter of length. If you lay the rope around the tennis ball in a circle again, you will get a distance of 16 cm between the rope and the ball.

Now consider the same for the earth. Assuming a perfect globe with an equator of 40 000 km and an additional meter for the rope: What will the distance between earth and the rope be?

##### Tennis ball (3.35 cm radius):

U1 = 2 × 0.0335 × π
U2 = U1 + 1
r2 = U2 / 2π
r2 = ~0.19265 m
r2 – r1 = 0.19265 – 0.0335 = +0.15915m

##### Globe (40 000 m radius):

U1 = 2 × 40000 × π
U2 = U1 + 1
r2 = U2 / 2π
r2 = ~40000.1591 m
r2 – r1 = 40000.1591 – 40000 = +0.15915m

##### Conclusion

The distance is always about 16cm for an additional meter. This comes from the formula `r = U / 2π`. You can separate the constant from the variables: `r2 = U1 / 2π + 1 / 2π`. So the 16cm is actually `parameter / 2π`.