Problem 23 · 1986 AJHSME
Stretch
Geometry & Measurement
area-differencehalf-circles
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Answer: B — 1.
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Hint 1 of 3
First nail down the sizes. Two small circles of radius 1 sit side by side along the diameter AC, so AC = 4, making the big circle's radius 2 β the big radius is double the small one.
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Hint 2 of 3
The shaded blob is the *top half* of the big circle with the *top halves* of the two small circles scooped out. Write shaded = (half the big disk) β (two half small disks) and let the Ο's do the work.
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Hint 3 of 3
Don't fear Ο β it will cancel when you form the final ratio, so the answer is a clean number.
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Approach: shaded = half big disk β two half small disks
- Sizes first: the two radius-1 circles span AC, so AC = 4 and the big radius is 2. Doubling the radius is the key β area grows with the *square* of the radius.
- The shaded region is the upper half of the big circle minus the upper halves of the two small circles: Β½(Ο Β· 2Β²) β 2 Β· [Β½(Ο Β· 1Β²)] = 2Ο β Ο = Ο.
- The question asks for the ratio of this to *one* small circle (area Ο Β· 1Β² = Ο): Ο β Ο = 1. The Ο cancels, leaving a tidy whole number.
- Neat takeaway: even though the big circle has 4Γ the area of a small one, slicing everything in half and subtracting leaves the shaded piece exactly equal to a whole small circle β a reminder that 'radius doubles β area quadruples' drives these comparisons.
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