### 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.

# Bobbly Perimeter

##### Age 14 to 16 Short Challenge Level:
Finding the lengths of the arcs
If the perimeter of the square is 20 cm, then each side must be 5 cm long.

The sides of the square are the diameters of the semicircles, so the circumferences of the full circles would be $5\times\pi=5\pi$ cm.

As shown in the diagram below, the 4 semicircles make up 2 full circles.

So the total perimeter is $5\pi+5\pi=10\pi$ cm.

Using scale factors
The sides of the square are the diameters of the semicircles, and so the circumferences of the full circles would be $\pi\times\text{diameter}=\pi\times\text{side length}$.

Each semicircle has only half the circumference of a full circle, so its length is $\frac{1}{2}\pi\times\text{side length}$.

So to go from a square to a semicircle, each side length is mutliplied by a scale factor of $\frac{1}{2}\pi$. So the perimeter must also be multiplied by this scale factor. So the perimeter of the new shape will be $20\times\frac{1}{2}\pi=10\pi$ cm.

You can find more short problems, arranged by curriculum topic, in our short problems collection.