Problem 5 · AMC 8 Stretch
Core
Geometry & Measurement
considering-extreme-cases
Two circles share the same center. The gap between them (big radius minus small radius) is \(10\). How much bigger is the big circle's circumference than the small circle's?
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Answer: \(20\pi\)
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Hint 1 of 3
The actual radii are never given, so the answer must depend only on the gap of \(10\). That is a hint to try an extreme case.
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Hint 2 of 3
Shrink the small circle to a single point. Then the gap, \(10\), IS the radius of the one circle that is left.
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Hint 3 of 3
A point has circumference \(0\). So the difference is just the circumference of a circle of radius \(10\). Compute \(2\pi \times 10\).
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Approach: Considering an extreme case — shrink the inner circle to a point
- Extreme case: shrink the inner circle to a point. Then the gap \(10\) becomes simply the radius of the surviving circle. Its circumference is \(2\pi(10) = 20\pi\), and the point has circumference \(0\), so the difference is \(20\pi\).
- Why this always works: the difference of circumferences is \(2\pi R - 2\pi r = 2\pi(R - r) = 2\pi(10) = 20\pi\), which depends only on the gap \(R-r\), not on the individual sizes.
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