Problem 16 · 2015 Math Kangaroo
Hard
Geometry & Measurement
area
The diagram shows three concentric circles and two perpendicular, common diameters of the three circles. The three grey sections are of equal area, the small circle has radius 1. What is the product of the radii of the three circles?
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Answer: A — √6
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Hint 1 of 2
Write each shaded region's area in terms of the three radii and set them equal.
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Hint 2 of 2
Work outward from the small circle of radius 1.
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Approach: set the three equal-area conditions and solve for the radii
- The three grey pieces (a quarter of the inner disk, and quarter-annulus bands of the middle and outer rings) have equal areas.
- Equal areas force the radii to satisfy r₁² = r₂²−r₁² = r₃²−r₂²; with r₁ = 1 this gives r₂² = 2 and r₃² = 3.
- So the product r₁·r₂·r₃ = 1·√2·√3 = √6 (A).
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