Problem 6 · 2024 Math Kangaroo
Medium
Geometry & Measurement
areaarea-decomposition
The diagram shows a square containing four touching circles of equal size. What is the ratio of the area of the black part to the area of the grey part?
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Answer: B — 1 : 3
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Hint 1 of 2
Pick an easy size: let each circle have radius 1, so the square has side 4.
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Hint 2 of 2
Cut the square into nine equal small squares; the black centre and the grey corners are made of those small squares minus the circle pieces inside them.
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Approach: split the leftover area into a centre square and the corners
- Let each circle have radius 1; then the square has side 4 and area 16, and the four circles cover \(4\pi\).
- Look at the central \(2\times 2\) square (area 4): it contains exactly four quarter-circles, so the black centre is \(4-\pi\).
- The grey region is everything else outside the circles, which equals \(16-4\pi-(4-\pi)=12-3\pi=3(4-\pi)\), exactly three times the black centre.
- So black : grey \(=1:3\), answer B.
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