Problem 25 · 2011 AMC 8
Hard
Geometry & Measurement
inscribed-circumscribedapproximation
A circle with radius 1 is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?
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Answer: A — Closest to 1/2.
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Hint 1 of 2
The two key fits: the circle sits inside the outer square, so the outer square's side equals the diameter; the inner square sits inside the circle with its corners on it, so the inner square's diagonal equals the diameter. Both are pinned to the diameter = 2.
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Hint 2 of 2
From a square's diagonal you get its area fast: area = (diagonal)2÷2. So the inner square's area is 22÷2 = 2 — no need for the √2 side length at all.
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Approach: tie every length to the diameter, then form the ratio
- Diameter = 2. Outer square (circle inscribed in it): side 2, area 4. Inner square (inscribed in the circle): diagonal 2, so area = 22÷2 = 2.
- Shaded part = inside the circle but outside the inner square = π(1)2 − 2 = π − 2.
- Area between the squares = outer − inner = 4 − 2 = 2.
- Ratio = (π − 2)÷2 ≈ (3.14 − 2)÷2 ≈ 0.57, closest to 1/2.
- Sanity check: π − 2 ≈ 1.14 is just over half of 2, so the ratio is just over 1/2 — 1/2 is clearly the nearest choice (the next option, 1, would need the shaded area to equal the full gap of 2).
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