Question

A square is inscribed in a circle of diameter ‘D’. If one side of the blue square is the diagonal of the red square, what is the ratio of the area of the smaller square and the circle?

• A. $2:\pi$
• B. $1:\pi$
• C. $D:\pi$
• D. $D:2\pi$

Answer 1

The correct option is B $1:\pi$

Let the length of the side of blue square and red square be 'a' and 'b'.
Length of diagonal of blue square $=\sqrt{2}a$

Here, diagonal of blue square = Diameter of the circle
$\Rightarrow \sqrt{2}a=D\Rightarrow a=\frac{D}{\sqrt{2}}$

And, diagonal of the red square = side of blue square
$\Rightarrow \sqrt{2}b=a\Rightarrow b=\frac{a}{\sqrt{2}}\Rightarrow b=\frac{D}{2}$

So, red square is a smaller square.
Area of red square $=b^2=(\frac{D}{2}{)}^2=\frac{D^2}{4}\text{sq. units}$
And area of circle
$=\pi (\frac{D}{2}{)}^2=\frac{\pi D^2}{4}\text{sq. units}$

Ratio of area of smaller sqaure to area of the circle
$=\frac{\frac{D^2}{4}}{\frac{\pi D^2}{4}}=\frac{1}{\pi }=1:\pi$