Question

The radius of a circle is 10 cm. If the radius is increased by 20%, how do you find the percentage increase in the area?

Answer 1

Hint: Initial radius is given as 10 cm and it is increased by 20%. First find the value of the increased radius. Correspondingly find the value of initial and final areas using formula $A = \pi {r^2}$. Then apply the formula of percentage increase, which is ${\text{Percentage Increase}} = \dfrac{{{\text{Final value}} - {\text{Initial value}}}}{{{\text{Initial value}}}} \times 100\% $, to calculate the final answer.

Complete step by step answer:
According to the question, we have been given the value of radius of a circle and it is said that radius is increasing by 20%. We have to find a way to calculate the percentage increase in the area.
The radius of the circle is given as 10 cm. So we have:
\[ r = 10{\text{ cm}}\]
We know that the formula for the area of circle is:
$ \Rightarrow A = \pi {r^2}$
Putting the value of radius in this formula, we’ll get the value of the initial area of the circle. Thus the initial area of the circle is:
$
   A = \pi {\left( {10} \right)^2} \\
   \Rightarrow A = 100\pi {\text{ }}.....{\text{(1)}}
 $
Now the radius is increased by 20%. Let the increased radius is denoted by ${r_1}$ then we have:
\[
   {r_1} = r + \dfrac{{20}}{{100}}r \\
   \Rightarrow {r_1} = r + \dfrac{1}{5}r \\
   \Rightarrow {r_1} = \dfrac{6}{5}r \\
 \]
Putting the value of initial radius \[r = 10{\text{ cm}}\], we’ll get:
\[
   {r_1} = \dfrac{6}{5} \times 10 \\
   \Rightarrow {r_1} = 12{\text{ cm}}
 \]
Therefore the radius is increased to 12 cm. Now let the increased area be denoted by ${A_1}$. Then we have:
$ \Rightarrow {A_1} = \pi {r_1}^2$
Putting the value of ${r_1}$, we’ll get:

$
  {A_1} = \pi {\left( {12} \right)^2} \\
   \Rightarrow {A_1} = 144\pi {\text{ }}.....{\text{(2)}}
 $
Now, the percentage increase in the value of any quantity is given by the formula as shown:
$ \Rightarrow {\text{Percentage Increase}} = \dfrac{{{\text{Final value}} - {\text{Initial value}}}}{{{\text{Initial value}}}} \times 100\% $
For this question, the percentage increase in the area of the circle will be:
$ \Rightarrow {\text{Percentage Increase}} = \dfrac{{{A_1} - A}}{A} \times 100\% $
Putting the values of areas from equation (1) and (2), we’ll get:
$
   \Rightarrow {\text{Percentage Increase}} = \dfrac{{144\pi - 100\pi }}{{100\pi }} \times 100\% \\
   \Rightarrow {\text{Percentage Increase}} = \dfrac{{44\pi }}{{100\pi }} \times 100 \\
   \Rightarrow {\text{Percentage Increase}} = 44\%
 $

Thus, the percentage increase in the area of the circle is 44%.

Note: If the percentage increase in the radius is given, we don’t necessarily have to calculate the initial and final areas of the circle. In the above problem, the initial area is $r$ and the final area is \[\dfrac{6}{5}r\] as we have already calculated. We can directly put these values in the percentage increase formula without calculating the exact numerical value. This is shown below:
$
   \Rightarrow {\text{Percentage Increase}} = \dfrac{{{A_1} - A}}{A} \times 100\% \\
   \Rightarrow {\text{Percentage Increase}} = \dfrac{{\pi {r_1}^2 - \pi {r^2}}}{{\pi {r^2}}} \times 100\% \\
 $
We have calculated the final radius and it is \[\dfrac{6}{5}r\], so we’ll get:
\[
   \Rightarrow {\text{Percentage Increase}} = \dfrac{{\pi {{\left( {\dfrac{6}{5}r} \right)}^2} - \pi {r^2}}}{{\pi {r^2}}} \times 100\% \\
   \Rightarrow {\text{Percentage Increase}} = \dfrac{{\dfrac{{36}}{{25}}\pi {r^2} - \pi {r^2}}}{{\pi {r^2}}} \times 100 \\
   \Rightarrow {\text{Percentage Increase}} = \left( {\dfrac{{36}}{{25}} - 1} \right) \times 100 \\
 \]
If we calculate this, we’ll get 44% which is our correct answer. Thus this method can also be used to solve the problems of these types.