# The ratio of the areas of incircle and circumcircle of a square is

$\\ \text{Let the side of a square be denoted as "a"}\\ \text{Radius of incircle is half the length of side of a square} =\frac{a}{2} \\ \text{Area of incircle} A_{1}=\pi\left(\frac{a}{2}\right)^{2}=\frac{\pi a^{2}}{4}\\ \text{Radius of circumcircle is half the length of the diagonal of square} =\frac{a}{\sqrt{2}} \\ \text{Area of circumcircle} A_{2}=\pi\left(\frac{a}{\sqrt{2}}\right)^{2}=\frac{\pi a^{2}}{2}\\ \therefore \frac{A_{1}}{A_{2}}=\frac{\frac{\pi a^{2}}{4}}{\frac{\pi a^{2}}{2}} =\frac{1}{2} =1: 2$

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