Problem 15 · 2018 AMC 8
Medium
Geometry & Measurement
areaarea-fraction
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Answer: D — 1 square unit.
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Hint 1 of 2
"Diameter of small = radius of big" secretly says the small circle's radius is half the big one's. And halving a radius doesn't halve the area — notice it does something stronger.
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Hint 2 of 2
The technique: area scales as the square of the radius. Half the radius means one-quarter the area. Once you have that, you never need π or the actual radius — just compare.
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Approach: areas scale as radius squared
- Each small radius is half the big radius, so each small area is (1/2)2 = 1/4 of the big circle's area. Two smalls together are 2×(1/4) = 1/2 of the big.
- We're told the two smalls total 1 square unit, and that's half the big, so the big circle is 2 square units.
- Shaded = big − the two white smalls = 2 − 1 = 1.
- You'll see it again: the "half the radius ⇒ quarter the area" jump (and its cousin "triple radius ⇒ nine times area") lets you compare circles without ever touching π.
Mark:
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