Problem 4 · 2021 Math Kangaroo
Medium
Geometry & Measurement
area-fractionarea
A large square is divided into smaller squares, as shown. A shaded circle is inscribed inside each of the smaller squares. What proportion of the area of the large square is shaded?
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Answer: E — \(\tfrac{\pi}{4}\)
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Hint 1 of 2
A circle inscribed in a square always fills the same fraction of that square.
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Hint 2 of 2
Find that fraction once; it does not depend on the square's size.
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Approach: use the fixed circle-to-square area ratio
- A circle inscribed in a square of side s has area π(s/2)² = πs²/4.
- That is π/4 of the square's area s², the same fraction for every square.
- Since the circles fill squares that tile the big square, the shaded part is π/4 of the whole.
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