Problem 7 · 2006 AMC 8
Easy
Geometry & Measurement
circle-formulas
Circle X has a radius of π. Circle Y has a circumference of 8π. Circle Z has an area of 9π. List the circles in order from smallest to largest radius.
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Answer: B — Z, X, Y.
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Hint 1 of 2
The circles are described in three different languages — radius, circumference, area. Translate all three into the SAME thing (radius) before you can compare them.
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Hint 2 of 2
Peel the radius out of each formula: C = 2πr and A = πr2. The lone π factors cancel, leaving clean whole-number radii.
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Approach: convert all three descriptions to radius
- Y: C = 2πr = 8π, so r = 4. Z: πr2 = 9π, so r2 = 9 and r = 3. X: r = π ≈ 3.14 (given directly).
- Now they're comparable: 3 < 3.14 < 4, so smallest to largest is Z, X, Y.
- The key fact for the close call: π lands between 3 and 4 (it's about 3.14), so circle X squeezes in between Z and Y. Anytime quantities are given in different forms, reduce them to one common measure first.
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