### At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

### Contact

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

### Gift of Gems

Four jewellers share their stock. Can you work out the relative values of their gems?

# Arrh!

##### Age 14 to 16 Challenge Level:

Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC.

What is the value of ${\bf \frac{r}{R}}$?