Question

If the radius of a circle is increased to twice its original length, how much will the area of the circle increase?

• A. $1.4$ times
• B. $2$ times
• C. $3$ times
• D. $4$ times

Answer 1

The correct option is D

$4$ times

Explanation for the correct option:

Step $1:$ Find the relation between both radii.

Assume that, the original radius of the circle be $r$.

Also assume that, the increased radius of the circle be $r_1$.

According to the given information.

$r_1=2r$.

Step $2:$Find the increase in the area of the circle.

Since the area of the original circle is ${\mathrm{\pi r}}^2$.

and the area of the circle after increasing the radius is $\mathrm{\pi }{\left({\mathrm{r}}_1\right)}^2$.

Therefore, the ratio of the two areas of the circle is: $\frac{\mathrm{\pi }{\left({\mathrm{r}}_1\right)}^2}{{\mathrm{\pi r}}^2}$.

$\Rightarrow \frac{\mathrm{\pi }{\left({\mathrm{r}}_1\right)}^2}{{\mathrm{\pi r}}^2}=\frac{{\left({\mathrm{r}}_1\right)}^2}{{\mathrm{r}}^2}$

Since, $r_1=2r$.

$$\begin{gathered} \Rightarrow \frac{\mathrm{\pi }{\left({\mathrm{r}}_1\right)}^2}{{\mathrm{\pi r}}^2}=\frac{{\left(2r\right)}^2}{{\mathrm{r}}^2} \\ \Rightarrow \frac{\mathrm{\pi }{\left({\mathrm{r}}_1\right)}^2}{{\mathrm{\pi r}}^2}=\frac{4r^2}{{\mathrm{r}}^2} \\ \Rightarrow \frac{\mathrm{\pi }{\left({\mathrm{r}}_1\right)}^2}{{\mathrm{\pi r}}^2}=\frac{4}{1} \end{gathered}$$

This means the area of the circle increases by $4$ times.

Hence, option $D$ is the correct option.