Problem 10 · 2010 Math Kangaroo
Medium
Geometry & Measurement
areapythagorean-triple
In the figure the square has side length 2. The semicircles pass through the midpoint of the square and have their centres on the corners of the square. The grey circles have their centres on the sides of the square and touch the semicircles. How big is the total area of the grey parts?
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Answer: A — \(4\cdot(3-2\sqrt{2})\cdot\pi\)
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Hint 1 of 2
Find the radius of a semicircle: its centre is a corner and it passes through the square's centre.
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Hint 2 of 2
A grey circle sits on a side and just touches a semicircle; relate their radii along that line.
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Approach: find both radii, then add four grey-circle areas
- A semicircle is centred at a corner and reaches the square's centre, a distance √2, so its radius is √2.
- A grey circle is centred at a side's midpoint, distance 1 from the nearest corner; touching the semicircle gives 1 + r = √2, so r = √2 − 1.
- One grey circle has area π(√2 − 1)² = π(3 − 2√2).
- Four of them total 4·(3 − 2√2)·π.
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