Problem 7 · 1997 AJHSME
Medium
Geometry & Measurement
inscribed-figure
The area of the smallest square that will contain a circle of radius 4 is
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Answer: D — 64.
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Hint 1 of 2
'Smallest square' means the circle just barely fits — it kisses all four sides at once. When that happens, what part of the circle exactly spans the square?
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Hint 2 of 2
A circle touching opposite sides of its tightest square sets the side = the circle's diameter (not its radius).
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Approach: the snug circle's diameter sets the side
- The smallest containing square has the circle tangent to all four sides, so the circle's full width — its diameter — equals the side. With radius 4, that's 2 × 4 = 8.
- Area = side² = 8² = 64.
- Trap: it's tempting to use the radius (4) as the side and get 16 — but the radius only reaches halfway across. Always picture the diameter spanning the gap.
- You'll see it again: 'tightest box around a circle' → side = diameter; 'tightest circle around a square' → diameter = the square's diagonal.
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