Question

Find the ratio of areas of inscribed square in a semi-circle and a circle while the radii of circle and semi-circle are equal :

• A. 2 : 5
• B. 3 : 5
• C. 4 : 5
• D. 1 : 5

Answer 1

The correct option is A 2 : 5

Let the side of square formed in semi-circle be a units and radius of semi circle be r units
$\therefore r^2=a^2+\frac{a^2}{4}=\frac{5a^2}{4}$
$\therefore a^2=\frac{4r^2}{5}$
For square inscribed in circle
${\left(2r\right)}^2=2b^2$
$\Rightarrow 4r^2=2b^2$
$\Rightarrow b^2=2r^2$
$\therefore$ Required ratio$=\frac{\frac{4r^2}{5}}{2r^2}$
$=\frac{4r^2}{5}\times \frac{1}{2r^2}=\frac{2}{5}=2:5$