### Dividing the Field

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two trapeziums each of equal area. How could he do this?

### Two Circles

Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?

### Star Gazing

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

# Loo Roll Emergency

##### Age 14 to 16 Short Challenge Level:
The amount of toilet paper left will be proportional to the area of the paper on the cross-section of the roll. The cross sections of the two rolls are shown below.

On the new roll, this area is $\pi\times6^2-\pi\times2^2=36\pi-4\pi=32\pi$ cm$^2.$

On my roll, this area is $\pi\times3^2-\pi\times2^2=9\pi-4\pi = 5\pi$ cm$^2.$

So $\frac{5}{32}=15.625\%$ of the toilet paper is left.
You can find more short problems, arranged by curriculum topic, in our short problems collection.