Let’s say a tennis ball has a diameter of 6 cms and a basketball a diameter of 46 cms.
Let’s also say a car tyre has a diameter of 60 cms and a monster truck tyre a diameter of 100 cms.
Now, the unexpected bit:
If we were to wrap a piece of string exactly once around the tennis ball, how much more string would we need to do the same thing around the basketball?
In the same way, if we were to wrap a piece of string exactly once around the car tyre, how much more string would we need to do the same thing around the monster truck tyre?
Believe it or not, in both cases it’s 40π cms, or approximately 126 cms!
Our intuition doesn’t like it, but even if you were to wrap a string around the Earth and then a second string around the Earth 20 cms higher than ground level (which would increases the diameter of the circle formed by 40 cms), the difference in string length would still be 40π cms!
It feels deep in our bones like the increase in string length would be a whole lot more pronounced for the Earth-circling situation than the tyre-circling situation, yet it’s exactly the same, and here’s the maths that proves it:
Because c (circumference) = πd (diameter), whenever the radius increases by a length of x metres, the circumference will always be c = π(d + x) = πd + πx.
This means that when the diameter of a circle increases by a length of x centrimetres, the circumference is increased by πx centimetres — which is completely independent of the original circumference or radius of the circle in question!
And when we fill in the formula for the situations cited above, we’ve always got a πx cms = 40π cms ≈ 126 cms difference in circumference.