Question

If the radius of a circle is increased by 10%, then find the area of the new circle.
A. $121\pi {r^2}$
B. $12.1\pi {r^2}$
C. $1.21\pi {r^2}$
D. None of these

Answer 1

Hint: We will just assume the value r of radius of our old circle and then apply the increment of 10% in it. Now, we have the new radius and we will put in this radius in the formula of area of a circle.

Complete step-by-step answer:
Let us first discuss the formula of area of the circle:-
It is given by the formula: $A = \pi {r^2}$ , where r is the radius of the circle.
Let the radius of the initial circle be r units.
Now let us do an increment of 10% in the radius.
So, the new radius will increase by 10% of r that is \[\dfrac{{10}}{{100}} \times r = \dfrac{r}{{10}}\] units.
The new radius will be $r + \dfrac{r}{{10}}$ units.
Let us denote the new radius by r’.
Simplifying it by taking the LCM of this, we will get:-
$r' = \dfrac{{10r + r}}{{10}} = \dfrac{{11r}}{{10}}$ units.
Now, putting this value of radius in the formula of the area of a circle that is: $A = \pi {r^2}$.
So, $A = \pi {\left( {\dfrac{{11r}}{{10}}} \right)^2}$.
Simplifying it by opening the square:-
$ \Rightarrow A = \pi \times \dfrac{{11r}}{{10}} \times \dfrac{{11r}}{{10}}$
Simplifying it. We will get:-
$ \Rightarrow A = \dfrac{{121\pi {r^2}}}{{100}}$
This is equivalent to:-
$ \Rightarrow A = 1.21\pi {r^2}$

So, the correct answer is “Option C”.

Note: The student must first look at the options if it contains $\pi $ or not. If it does not, then we need to put in the value of it to get some answer among the options, otherwise you will have to keep the $\pi $ as it is.
The students should notice that here we do not have units in the options. This is done here because r is expressed in terms of units. So, the unit will itself appear when we put in the value of r.
Fun Facts about circle:- A circle is the only one sided shape with an area.
A straight line is a circle with an infinite area.
Circles have no angles.
A circle has an infinite amount of lines of symmetry.
The awesome word encyclopedia literally means "circle of learning".