Problem 25 · 2008 AMC 8
Hard
Geometry & Measurement
annulus-area
Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Approximately what percent of the design is black?
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Answer: A — About 42%.
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Hint 1 of 2
A ring (annulus) is just the big disk with the smaller disk punched out, so its area is πR² − πr² — never measure the ring directly.
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Hint 2 of 2
You want a ratio, so the π will cancel — work with the radius-squares and only deal with whole numbers.
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Approach: rings as differences of disks, then take the ratio (π cancels)
- The radii are 2, 4, 6, 8, 10, 12. Each black region is a ring = outer disk − inner disk, so use π(R² − r²): the center black disk is π·2² = 4π, the ring from 4 to 6 is π(36 − 16) = 20π, and the ring from 8 to 10 is π(100 − 64) = 36π.
- Black total: 4π + 20π + 36π = 60π; whole design: π·12² = 144π.
- Ratio: 60π / 144π = 60/144 = 5/12 ≈ 41.7%, closest to 42%. The π cancels, so you're just comparing radius-squares.
- Why this transfers: any "ring" or "washer" area is one disk minus another, and in a fraction-of-the-whole question every π divides out — track only the squared radii.
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