Question

A semi-circle is inscribed in a quarter circle as shown in the figure,

What fraction of the quarter circle is shaded ?

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{3}{4}$
• D. $\frac{3}{5}$

Answer 1

The correct option is B $\frac{2}{3}$

Let the radius of quarter circle be $R$ and semicircle be $r$
Area fraction can be written as,
Area of semi-circle $A_1=\frac{1}{2}\pi r^2$
Area quarter circle $A_2=\frac{1}{4}\pi R^2$
$\therefore \frac{A_1}{A_2}=\frac{2r^2}{R^2}$
Constructing a square as $OACD$ as shown,

using Pythagoras theorem in $△OCE$
$R^2=r^2+(r\sqrt{2}{)}^2\Rightarrow R^2=3r^2$

Hence the ratio will be
$\frac{A_1}{A_2}=\frac{2}{3}$