Problem 19 · 2023 Math Kangaroo
Hard
Geometry & Measurement
areaarea-decomposition
A square with side length 30 cm is split into 9 squares. The big square contains three circles with radii 5 cm (bottom right), 4 cm (top left) as well as 3 cm (top right) as seen in the diagram. How many cm\(^2\) are shaded in grey?
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Answer: B — \(500\)
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Hint 1 of 2
The 30 cm square splits into nine 10 cm squares; work out which of those are shaded grey.
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Hint 2 of 2
Add up the grey square areas and account for the circle pieces that fall in them.
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Approach: pair each circular piece with an equal opposite piece so the \(\pi\) terms cancel
- The 30 cm square (area 900) is cut into nine 10×10 cells, each of area 100.
- The shading takes some whole cells and then swaps circle pieces in and out: for each circle, the grey part removed from one cell is matched by an equal circular part added in an adjacent cell.
- Because every circle contributes the same amount of grey as it removes, all the \(\pi\) terms cancel and the grey area is a whole number of square cells.
- Adding up the net grey cells gives 500 cm² (choice B); the \(\pi\)-containing options are traps for anyone who forgets the pieces cancel.
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