Rolling inside
Weekly Problem 11 - 2007
A circle of radius 1 rolls without slipping round the inside of a square of side length 4. Find an expression for the number of revolutions the circle makes.
A circle of radius 1 rolls without slipping round the inside of a square of side length 4. Find an expression for the number of revolutions the circle makes.
Age
14 to 16
Challenge level
Problem
A circle of radius 1 rolls without slipping round the inside of a square of side length 4. Find an expression involving $\pi$ for the exact number of revolutions the circle makes by the time it returns to its original position.
Image
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
$4/\pi$
The circumference of the circle is $2\pi$. This is the distance its centre moves each time the circle rolls for one revolution. When the circle moves from one corner to an adjacent corner, its centre moves a distance 2, so the circle makes $1/\pi$ revolutions. As it needs to do this four times before the circle returns to its original position, the number of revolutions is $4/\pi$.