Discussion
What is ratio of areas of circumcircle and in circle of a square?
Answer: D
For a square of side \(a\), the circumcircle passes through the vertices and has the diagonal as its diameter, and the incircle touches the sides and has the side as its diameter.
\[ R = \frac{a\sqrt{2}}{2},\qquad r = \frac{a}{2} \]
\[ \frac{\pi R^2}{\pi r^2} = \frac{a^2/2}{a^2/4} = 2 \]
So the areas are in the ratio 2 : 1. The ratio of the radii is \(\sqrt{2} : 1\), so the ratio of the areas is the square of that. The ratios 1 : 2 and 1 : 3 are not correct, and 3 : 1 is not either. The answer is 2 : 1, option D.
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