Question

If the circumference of a circle increases \[4\pi \]to \[8\pi \], what changes occur in its area?
A) It is halved.
B) It doubled.
C) It triples
D) It quadruples

Answer 1

 Hint- To find the change in the area of the circle we need to find the radius first. With help of the change in circumference we will calculate the change of radius.
We know that the circumference of a circle of radius \[r\] is \[2\pi r\].
We know that the area of a circle of radius \[r\] is \[\pi {r^2}\].

Complete step by step answer:
To find the change in the area of the circle we need to find the radius first. With help of the change in circumference we will calculate the change of radius.
We know that the circumference of a circle of radius \[r\] is \[2\pi r\].
Let us consider the radius of the circle to be \[r\].
According to the problem,
\[2\pi r = 4\pi \]
Solving we get, the radius \[r = 2\]
Again, let us consider the radius of the circle changes \[r\]to \[R\]when the circumference changes into \[4\pi \]to \[8\pi \]
Then according to the problem,
\[2\pi r = 8\pi \]
Solving we get, the radius \[R = 4\]
So, the radius increases \[2\]to \[4\].
Now we will find the area of the circle.
We know that the area of a circle of radius \[r\] is \[\pi {r^2}\].
When the radius is \[r = 2\], the area is,
\[{A_1} = \pi {(2)^2} = 4\pi \]
Again,
When the radius is \[r = 4\], the area is,
\[{A_2} = \pi {(4)^2} = 16\pi \]
Now,
\[\dfrac{{{A_1}}}{{{A_2}}} = \dfrac{{4\pi }}{{16\pi }} = \dfrac{1}{4}\]
So, we get,
\[{A_2} = 4{A_1}\]
Hence,
When the circumference of a circle increases \[4\pi \]to \[8\pi \], the area of the circle quadruples.
The correct option (D) it quadruples.

Note – If the circumference of a circle increases or decreases, the radius will change according to that. Again if the radius of a circle increases or decreases, the area of the circle will be changing according to that.