### Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

### Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..

### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

# Contact

##### Stage: 4 Challenge Level:

Think about a coin with radius $1$ unit in the corner of a box. I roll the coin along the side of the box, making sure it doesn't slip.

How long would the box need to be for every point on the coin's circumference to touch the box as I roll it along?

Now imagine the coin rolling along two edges of a box like this:

If the two edges are the same length, how long do they need to be to ensure every point on the circumference touches? What if they're not the same length?

 Now imagine rolling the coin around on a rectangular tray so that the edge of the coin is always in contact with the inside wall of the tray, and so that it always rolls and never slides. What proportion of the circumference touches the tray as it makes one complete circuit of a $4$ by $3$ tray? Investigate with the interactivity below for different sizes of tray.

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Can you draw any diagrams to help to explain your thinking?

What is the smallest tray you can find where all of the circumference touches the tray at some point?