Question

How does doubling the radius of a circle affect its area?

Answer 1

Hint: For this problem we will assume a circle with radius $r$. Now we will calculate the area of the circle which is having the radius $r$ and we will denote it as ${{A}_{1}}$. In the problem they have mentioned that the radius is doubled, so now the radius of the circle will be changed to $2r$. Now we will calculate the area of the circle which is having radius $2r$ and denote it as ${{A}_{2}}$. To get the result we will establish a relation between the calculated areas ${{A}_{1}}$ and ${{A}_{2}}$.

Complete step-by-step answer:
Let us assume a circle having the radius $r$ which is shown in below figure.

Now the area of the circle having the radius $r$ is given by
${{A}_{1}}=\pi {{r}^{2}}$
In the problem they have mentioned that the radius of the circle is doubled. Then the radius of the circle is changed as $2r$ and the circle is shown in below.

Now the area of the above circle is given by
$\begin{align}
  & {{A}_{2}}=\pi {{\left( 2r \right)}^{2}} \\
 & \Rightarrow {{A}_{2}}=\pi \left( 4{{r}^{2}} \right) \\
 & \Rightarrow {{A}_{2}}=4\pi {{r}^{2}} \\
\end{align}$
We have the area of the circle having the radius $r$ is ${{A}_{1}}=\pi {{r}^{2}}$. Substituting this value in the above equation, then we will get
$\therefore {{A}_{2}}=4{{A}_{1}}$
Hence, we can say that the area of the circle will be increased by $4$ times when the radius is doubled.

Note: In the problem they have mentioned that the radius is doubled. If they have mentioned the radius is increased by $3$ times or $4$ times. Then also we need to follow the above procedure. But the above procedure is somewhat lengthy, so we have the shorter method i.e., when the radius of the circle is increased by $n$ times, then the area will be increased by ${{n}^{2}}$ times.