Problem 12 · 2024 Math Kangaroo
Hard
Geometry & Measurement
area-decomposition
In a square with side length 6, a diagonal, a semicircle and a quarter circle are drawn as shown. What is the area of the grey region?
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Answer: A — 9
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Hint 1 of 3
Don't compute each curved piece separately — look for curved areas that exactly cancel.
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Hint 2 of 3
The diagonal pairs the semicircle and quarter-circle so their \(\pi\) contributions cancel, leaving a whole-number area.
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Hint 3 of 3
Express the grey region as a triangle plus/minus curved pieces and watch the \(\pi\)-terms vanish.
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Approach: pair the curved pieces so the \(\pi\)-terms cancel
- The square has area \(6^2=36\) and the diagonal splits it into two right triangles of area \(18\).
- The grey region is a triangular part with one curved bite added and an equal curved bite removed: the semicircle and quarter-circle pieces contribute opposite \(\pi\)-areas.
- Those circular contributions cancel exactly, so the grey area is a clean whole number.
- Evaluating, the grey region has area 9 (answer A).
Mark:
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