Question

In the figure given above, $\overline{AC}$ is a diameter of the large circle and B lies on $\overline{AC}$ so that $\overline{AB}$ is a diameter of the small circle. If $AB=1$ and $BC=2$, Calculate the area of the shaded region.

• A. $\frac{\pi }{4}$
• B. $\pi$
• C. $2\pi$
• D. $\frac{9\pi }{4}$
• E. $9\pi$

Answer 1

The correct option is C $2\pi$

To find the area of the shaded region, we have to find the area of the bigger circle and the area of the smaller circle.
For large circle, $AC$ is diameter measuring $3$.
$AC=AB+BC=1+2=3$
Area of larger circle is $=$ $\pi R^2$
$\pi {\left(\frac{3}{2}\right)}^2=\pi \frac{9}{4}$
Area of smaller circle is $=\pi r^2$
$\pi {\left(\frac{1}{2}\right)}^2=\pi \frac{1}{4}$
Area of shaded region is $=$ Area of larger circle $-$ Area of smaller circle
$=\frac{9\pi }{4}-\frac{\pi }{4}$
$=\frac{9\pi -\pi }{4}=\frac{8\pi }{4}=2\pi$