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Four semicircles are drawn on a line to form a shape. What is the area of this shape?
Age
14 to 16
Challenge level
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?
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If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
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 $$\left(\frac{1}{2}\times\pi\times 3^2+\frac{1}{2}\times\pi\times 1^2\right)\;\mathrm{cm}^2 = 5\pi\;\mathrm{cm}^2\;.$$