Smile
Four semicircles are drawn on a line to form a shape. What is the area of this shape?
Problem
The points $P$, $Q$, $R$, $S$ lie in order along a straight line, with $PQ=QR=RS=2\;\mathrm{cm}$. Semicircles with diameters $PQ$, $QR$, $RS$ and $SP$ join to make the shape shown below.
What is the area of the shape?
Image
![Smile Smile](/sites/default/files/styles/large/public/thumbnails/content-id-10200-15_weekly52.png?itok=XYYYABmA)
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
If the semicircle with diameter $PQ$ is rotated through $180^\circ$ about $Q$, the new shape formed has the same area as the original shape. It consists of a semicircle of diameter $6\;\mathrm{cm}$ and a semicircle of diameter $2\;\mathrm{cm}$.
So the area of the original shape is