Question

In the above figure, a square is inscribed in a circle of diameter $d$ and another square is circumscribing the circle. Find the ratio of the area of the outer square to the area of the inner square.

Answer 1

The side of the big square is equal to the diameter of the circle and the length of the diagonal of the small square is equal to the diameter of the circle.

Let $b_1$ be the side of the big square and $b_2$ be the side of the small square.

So, we can say that $b_1=d$.

And,

$b_2^2+b_2^2=d^2$

$\Rightarrow \sqrt{2}{b}_2=d$

$\Rightarrow b_2=\frac{d}{\sqrt{2}}$

Now the ratio of the area of two squares will be,

$\begin{array}{cc}\text{Ratio of area}& =\frac{b_1^2}{b_2^2}\\ & =\frac{d^2}{{\left(\frac{d}{\sqrt{2}}\right)}^2}\\ & =\frac{d^{2}2}{d^2}\\ & =2\end{array}$