# 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.

## 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.

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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 $4/\pi$.