Question

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

Answer 1

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 =\[\pi \left( \frac{a}{\sqrt{2}} \right)^2\]       .....(i)
Now, the radius of the circle inscribed in a square =\[\frac{a}{2}\]

Hence, area of the circle inscribed in a square = \[\pi \left( \frac{a}{2} \right)^2\]           .....(ii) 

From (i) and (ii) 

\[\frac{\text{ Area of circle circumscribing a square }}{\text{ Area of circle inscribed in a square }} = \frac{\pi \left( \frac{a}{\sqrt{2}} \right)^2}{\pi \left( \frac{a}{2} \right)^2}\]
\[ = \frac{\frac{1}{2}}{\frac{1}{4}}\]
\[ = \frac{2}{1}\]

Hence, the required ratio is 2 : 1.