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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
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
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 $$\left(\frac{1}{2}\times\pi\times 3^2+\frac{1}{2}\times\pi\times 1^2\right)\;\mathrm{cm}^2 = 5\pi\;\mathrm{cm}^2\;.$$