Question

The area of the square inscribed in a semicircle is $2c{m}^2$ . If the semi-circle is extended to a full circle, the area of the square inscribed in this circle is

• A. $6c{m}^2$
• B. $7c{m}^2$
• C. $4c{m}^2$
• D. $5c{m}^2$

Answer 1

The correct option is D $5c{m}^2$


Area of square = $2c{m}^2$
Side of square = $\sqrt{2}cm$
Let 'r' be the radius of the semicircle.
Then, from the figure,
$r^2=(\sqrt{2}{)}^2+{\left(\frac{\sqrt{2}}{2}\right)}^2\Rightarrow r^2=2+\left(\frac{2}{4}\right)\Rightarrow r^2=\frac{5}{2}\Rightarrow r=\sqrt{\left(\frac{5}{2}\right)}cm$


When the semicircle is extended to form a full circle, the diagonal of the square inscribed in it will be the diameter of the circle.
$\text{Diameter}=2\times \sqrt{\left(\frac{5}{2}\right)}\text{Diagonal of square}=\text{side}\times \sqrt{2}\Rightarrow \text{side}\times \sqrt{2}=2\times \sqrt{\left(\frac{5}{2}\right)}\Rightarrow \text{side}\times 2=2\times \sqrt{5}\Rightarrow \text{side}=\sqrt{5}cm$

$\therefore \text{Area of square}={\text{side}}^2=(\sqrt{5}{)}^2=5c{m}^2$