Question

Find the ratio of the area of the circle circumscribing a square to the area of the circle inscribed in the square .

Solution

Let the side of the square inscribed in a square be a units. Diameter of the circle outside the square = Diagonal of the square = $\sqrt{2}a$ Radius = $\frac{\sqrt{2}a}{2}=\frac{a}{\sqrt{2}}$                               So, the area of the circle circumscribing the square = $\mathrm{\pi }{\left(\frac{\mathrm{a}}{\sqrt{2}}\right)}^{2}$                         .....(i) Now, the radius of the circle inscribed in a square = $\frac{a}{2}$ Hence, area of the circle inscribed in a square = $\mathrm{\pi }{\left(\frac{\mathrm{a}}{2}\right)}^{2}$                                     .....(ii) From (i) and (ii) Hence, the required ratio is 2 : 1.  MathematicsRD Sharma (2016)Standard X

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