Problem 13 · 2018 Math Kangaroo
Hard
Geometry & Measurement
area-decompositionarea
Eight congruent semicircles are drawn inside a square with side length 4. How big is the area of the white part?
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Answer: B — 8
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Hint 1 of 2
Find the radius of the semicircles from how they sit along the sides.
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Hint 2 of 2
Pair up the grey and white pieces so the curved parts cancel.
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Approach: match curved pieces so the area is exactly half the square
- Two semicircles fit along each side of length 4, so each has diameter 2 and radius 1.
- By symmetry the eight semicircles split the square into equal grey and white regions: each curved bite removed from the grey is matched by an equal curved bite added to the white.
- So the white area is exactly half of the square: \(\tfrac12\times 16 = \) 8.
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