Radius is always proportional to circumference, so a circle twice the size is twice as big around.
Take the case of two identical circles. To move a point on the first circle from 12 o'clock back to 12 o'clock, it only goes halfway around the other circle, which you can prove to yourself by imagining you've wrapped a string around the circle and marked it at 12 and 6 o'clock. If you unwrap half of the string and wrap it around the other circle, then the end of the rope is at 6 o'clock. To roll the string back up by moving the circle, the top of the circle will be pointing upward again when it reaches the bottom. 1 full revolution. Now wrap the other half of the string around the other side, 6 has to go back to the bottom again to roll the string back up. 2 revolutions.
It’s interesting to read others reasoning on this. I find it hard to follow and much harder to generalize without some notation (and a diagram which this comment box struggles to reproduce)
Radius is always proportional to circumference, so a circle twice the size is twice as big around.
Take the case of two identical circles. To move a point on the first circle from 12 o'clock back to 12 o'clock, it only goes halfway around the other circle, which you can prove to yourself by imagining you've wrapped a string around the circle and marked it at 12 and 6 o'clock. If you unwrap half of the string and wrap it around the other circle, then the end of the rope is at 6 o'clock. To roll the string back up by moving the circle, the top of the circle will be pointing upward again when it reaches the bottom. 1 full revolution. Now wrap the other half of the string around the other side, 6 has to go back to the bottom again to roll the string back up. 2 revolutions.