### 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.

# Semicircle Distance

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

Let the length of the rectangle be $x$. Then by considering the area of the whole rectangle, we can find $x$.

The area of the whole rectangle can be found using length $\times$ width = $10x$, as shown below.

This must be equal to the shaded area added to the total area of the two semicircles. The radius of each semicircle must be half of 10 cm, which is 5 cm.

So the red area is equal to $\pi\times 5^2$ cm$^2=25\pi$ cm$^2$, because the two red semicircles could be stuck together to make a circle of radius 5 cm.

That means that the total area of the rectangle must be $125 + 25\pi$ cm$^2$, so $10x=125+25\pi$, so $x=\dfrac{125+25\pi}{10}=12.5+2.5\pi$.

Now, as shown below, the shortest distance can be found using $x$.

$d=x-5-5=x-10$, so $d=12.5+2.5\pi-10=2.5+2.5\pi$ cm.

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