3. The diagram shows a circle with radius 1 that rolls without slipping around the inside of a square with sides of length 5 .
The circle rolls once around the square, returning to its starting point.
What distance does the centre of the circle travel?
A $16-2 \pi$
B 12
C $6+\pi$
D $20-2 \pi$
E 20
SOLUTION
B

As the circle rolls, its centre is always at a distance 1 from the square. Therefore, as shown in the diagram, the centre traces out a square whose side length is 2 less than the side length of the square. It follows that the centre travels a distance equal to the length of the perimeter of a square with side length 3 .
We deduce that the distance that the centre travels is 12 .