Problem 18 · 2025 AMC 8
Hard
Geometry & Measurement
areaarea-fraction
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Answer: B — R = 2.
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Hint 1 of 2
Don't compute either crescent-region's actual area. The two pictures are the same shape scaled up — so ask how area responds when you scale a figure.
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Hint 2 of 2
First translate the words into one number: "one quarter of the right region = the whole left region" means the right region is 4× the left. Since area scales as (length)2, the length scale-up is √4.
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Approach: similar figures: area scales as the square of the length
- Both pictures are the identical shape (a square inscribed in a circle), just enlarged — so every area on the right is R2 times the matching area on the left, because radius 1 scales up to R.
- The condition says a quarter of the right's between-region equals the left's whole between-region, so the right's whole region is 4 times the left's: R2 = 4.
- R = √4 = 2.
- Why this transfers: for similar figures, areas scale as length2 (and volumes as length3). Whenever shapes are merely scaled copies, you can skip the messy individual areas and just take a square root of the area ratio to get the length ratio.
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