Question

The ratio of the area of a square inscribed in a semi-circle to that of the area of a square inscribed in the circle of the same radius is:

• A. $2:1$
• B. $2:3$
• C. $2:5$
• D. $1:3$

Answer 1

Answer: (C)

$2:5$

To find:

$\frac{\text{Area of square}I}{\text{Area of square}II}=\frac{A_1}{A_1}=\frac{s^2}{a^2}$

Let the radius of the circle be $r$.

Referring to the figure, we get that

$s^2+\frac{s^2}{4}=r^2$ [Using Pythagorus theorem]

$\Rightarrow r=\frac{\sqrt{5}}{2}s$

Area of the square $I,A_1$= $s^2$

Now the other circle has same radius as $r$,

Therefore, The diameter of this circle is $2r=\sqrt{5}s$

The diameter is same as the length of diagonal of square $II$.

Let the length of square's side be $a$, then

$a^2+a^2=5s^2$

$\Rightarrow 2a^2=5s^2$

$\Rightarrow \frac{s^2}{a^2}=\frac{2}{5}$

Hence, the ration will be $2:5$