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Three semicircles showing the rectangle and parts of a circle
The shaded area may be divided into a $ 2 \times 1$ rectangle plus a semicircle plus two quarter circles
(all of radius $1$).

Hence the total area is that of the rectangle
plus a circle of radius $1$ which equals $2 + \pi$



This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.