Question

select the correct answer from the drop - down menu.
in the figure, oa = r and oc = r.
the sector cob is cut from the circle with center o. the ratio of the area of the sector removed from the whole circle to the area of the sector removed from the inner circle is
options: r^2/r^2, r/r, (r^2 - r^2)/r^2, 1

Explanation:

Step1: Define sector area formula

The area of a sector with central angle $\theta$ (in radians) in a circle of radius $r$ is $\frac{1}{2}r^2\theta$.

Step2: Find area of removed whole circle sector

For the whole circle (radius $R$), removed sector area: $A_{whole} = \frac{1}{2}R^2\theta$

Step3: Find area of removed inner circle sector

For the inner circle (radius $r$), removed sector area: $A_{inner} = \frac{1}{2}r^2\theta$

Step4: Calculate the ratio

Take the ratio $\frac{A_{whole}}{A_{inner}} = \frac{\frac{1}{2}R^2\theta}{\frac{1}{2}r^2\theta}$. The $\frac{1}{2}$ and $\theta$ terms cancel out.
$\frac{A_{whole}}{A_{inner}} = \frac{R^2}{r^2}$

Answer:

$\frac{R^2}{r^2}$ (matches the option $R^2/r^2$)