Problem 20 · 2014 AMC 8
Medium
Geometry & Measurement
quarter-circle-areapi-approximation
Rectangle ABCD has sides CD = 3 and DA = 5. A circle of radius 1 is centered at A, a circle of radius 2 is centered at B, and a circle of radius 3 is centered at C. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?
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Answer: B — 4.0.
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Hint 1 of 3
Each circle is centered at a corner of the rectangle — and a corner is a 90° right angle. What fraction of a full circle does a 90° wedge capture? That fraction of each circle is the only part inside the rectangle.
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Hint 2 of 3
So you don't subtract three whole circles — you subtract a quarter of each: area = rectangle − (¼)(sum of the three circle areas).
Still stuck? Show hint 3 →
Hint 3 of 3
The radii (1, 2, 3) all fit without the quarters overlapping, so just add πr2/4 for each.
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Approach: a corner captures exactly one quarter of each circle
- Each circle sits at a corner, where the two rectangle sides meet at 90°. A 90° angle is ¼ of the full 360°, so exactly one quarter of each circle pokes into the rectangle.
- Quarter areas: (¼)π(1)2 + (¼)π(2)2 + (¼)π(3)2 = (¼)π(1 + 4 + 9) = 14π/4 = 7π/2.
- Region outside all circles = rectangle − quarters = (3 × 5) − 7π/2 = 15 − 7π/2.
- 7π/2 ≈ 7(3.14)/2 ≈ 11.0, so the answer ≈ 15 − 11.0 = 4.0.
- Spot it next time: a circle centered at a polygon's corner always contributes a wedge equal to (corner angle ÷ 360°) of the circle — quarter at a square corner, third at an equilateral-triangle corner, and the three corner-angles of any triangle even sum to a half-circle.
Another way — estimate with π ≈ 22/7:
- Quarters total 7π/2. Using π ≈ 22/7: (7/2)(22/7) = 11 exactly.
- Rectangle 15 − 11 = 4.0 — the 22/7 estimate lands the multiple-choice answer with no decimals.
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