Problem 21 · 2010 Math Kangaroo
Hard
Geometry & Measurement
area-fraction
In the figure we see semicircles with radii 2 cm, 4 cm, or 8 cm. What fraction of the area is grey?
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Answer: B — ¼
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Hint 1 of 2
Semicircle areas grow with the square of the radius (2, 4, 8).
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Hint 2 of 2
Build the grey region from half-discs of radii 2, 4 and 8, then add and subtract.
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Approach: combine semicircle areas
- Take the big semicircle of radius 8 as the whole region; its area is ½π·8² = 32π.
- The S-shaped boundary (radius-4 and radius-2 semicircles) carves the grey lobe, whose area works out to 8π.
- Grey fraction = 8π / 32π = 1/4.
Mark:
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