How does doubling the diameter of a circle change its area?

Hint: Here, we will find the area of the circle with the given diameter and the area of the circle whose diameter is doubled by using the area of the circle formula. We will use both the area of the circles, to find the change in the area of the circle. Thus, the change in the area of a circle whose diameter is doubled is the required answer.

Formula Used:
We will use the following formula:
1. Radius is half the Diameter i.e.,\[r = \dfrac{d}{2}\]
2. Area of the circle is given by \[A = \pi {r^2}\] where \[r\] is the radius of the circle.

Complete step by step solution:
Let us consider a circle whose original diameter is \[d\].
We know that Radius is half the Diameter i.e.,\[r = \dfrac{d}{2}\]
We know that the Area of the circle is given by \[A = \pi {r^2}\] where \[r\] is the radius of the circle.
Now, by using the radius of the circle in the area of the circle formula, we get
\[ \Rightarrow A = \pi {\left( {\dfrac{d}{2}} \right)^2}\]
Applying the exponent on the terms, we get
\[ \Rightarrow A = \dfrac{{\pi {d^2}}}{4}\]………………………………………………………\[\left( 1 \right)\]

Now, let us consider the circle whose diameter is double the diameter of a circle.
So, we get \[{d_1} = 2d\]
We know that the Area of the circle is given by \[A = \pi {r^2}\] where \[r\] is the radius of the circle.
Now, by substituting \[r = \dfrac{{{d_1}}}{2}\] in the area of the circle formula, we get
\[{A_1} = \pi {\left( {\dfrac{{{d_1}}}{2}} \right)^2}\]
Substituting \[{d_1} = 2d\] in above equation, we get
\[ \Rightarrow {A_1} = \pi {\left( {\dfrac{{2d}}{2}} \right)^2}\]
Applying the exponent on the terms, we get
\[ \Rightarrow {A_1} = \dfrac{{4\pi {d^2}}}{4}\]
\[ \Rightarrow {A_1} = \pi {d^2}\]………………………………………………………\[\left( 2 \right)\]
Now, by substituting equation \[\left( 2 \right)\] in\[\left( 1 \right)\], we get
\[ \Rightarrow A = \dfrac{{\pi {d^2}}}{4}\]
\[ \Rightarrow A = \dfrac{{{A_1}}}{4}\]
Multiplying 4 on both the sides, we get
\[ \Rightarrow {A_1} = 4A\]

Therefore, if the diameter of the circle is doubled, then the area of the circle is four times larger than the area of the circle with diameter \[d\].

Note:
We know that Circle is a two dimensional curve whose distance from the center to any point on the curve is a constant which is called the radius of the circle. Area of a circle is the region occupied by a circle. We should remember that when finding the area of the circle, we will use the radius of the circle and not the diameter of the circle. We should remember that if the diameter is doubled, then the radius is the original diameter.