Question

If the radius of a given circle is increased by 50%, then find the increase in the area of that circle?
(a) 125%
(b) 100%
(c) 75%
(d) 50%

Answer 1

Hint: We first assume the radius and area of the circle before and after increasing the value of radius. We convert the new radius in terms of the old radius after the increment. Now, we use a new radius to find the new area of the circle. After finding the new area, we find the % increase in the area of the circle.

Complete step-by-step solution:
Given that we have a circle and we need to find an increase in the area if the radius of the circle is increased by $50\%$.
Let us assume the radius of the circle be ‘r’ and the area of the circle be ‘A’ before increasing the radius.
We know that area of a circle of radius ‘r’ is $A=\pi {{r}^{2}}$.
Before increasing the radius, we assume Area of the circle be $A=\pi {{r}^{2}}$ -----(1).
Now, we increase the radius by $50\%$. So, the final value of the radius will be $150\%$ of the given radius.
Let us assume the new radius to be ${{r}_{1}}$. So, we have \[{{r}_{1}}=150% of r\].
We know that x% of y = $\dfrac{x}{100}\times y$.
So, we have ${{r}_{1}}=\dfrac{150}{100}\times r$.
We now get ${{r}_{1}}=1.5r$ ---(2).
Now, we find the new area of the circle. Let us assume a new area ${{A}_{1}}$.
So, we have ${{A}_{1}}=\pi {{r}_{1}}^{2}$.
We use the value of ${{r}_{1}}$, that we got from equation (2).
${{A}_{1}}=\pi \times {{\left( 1.5r \right)}^{2}}$.
${{A}_{1}}=\pi \times \left( 2.25{{r}^{2}} \right)$.
${{A}_{1}}=2.25\times \left( \pi {{r}^{2}} \right)$.
From equation (1), we have
${{A}_{1}}=2.25\times A$.
${{A}_{1}}=2.25A$.
Let us find the percentage of the area increased. If x is increased to ${{x}_{1}}$, we know that percentage increase in x is calculated as $%change=\dfrac{{{x}_{1}}-x}{x}\times 100$.
% increase in area = $\dfrac{2.25A-A}{A}\times 100$.
% increase in area = $\dfrac{1.25A}{A}\times 100$.
% increase in area = $1.25\times 100$.
% increase in area = $125\%$.
$\therefore$ There is a $125\%$ increase in the area of the circle.
The correct option is (a).

Note: We need to calculate the percentage increase in area with respect to the original area not with respect to the new area. We should not directly square $50\%$ and say a $2500\%$ increase in the area of the circle. Similarly, we can get questions to find the percentage increase of the perimeter of the circle.