Problem 18 · 2025 Math Kangaroo
Hard
Geometry & Measurement
area-fractionarea
The area of the black semicircle shown is 12 cm². What is the area of the large quarter circle?
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Answer: D — 30 cm²
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Hint 1 of 3
Let the quarter circle have radius \(R\) and the small semicircle radius \(r\); read their relationship off the figure.
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Hint 2 of 3
The semicircle's diameter spans from the corner to the centre of the quarter circle's straight side, which forces \(R^2 = 5r^2\).
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Hint 3 of 3
Then just take the ratio of the two area formulas—the \(\pi\) cancels.
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Approach: write both areas with their radii and take the ratio
- Quarter circle: \(\tfrac14\pi R^2\); black semicircle: \(\tfrac12\pi r^2\).
- The figure fixes the semicircle so that \(R^2 = 5r^2\), hence the quarter circle is \(\dfrac{\tfrac14 R^2}{\tfrac12 r^2} = \dfrac{R^2}{2r^2} = \dfrac{5}{2}\) times the semicircle.
- So the quarter circle \(= \tfrac52 \times 12 = \textbf{30 cm}^2\), choice (D).
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