If the radius is doubled, how does the area of a circle change?

If the radius is doubled, how does the area of a circle change?

• A. The area remains the same
• B. The area doubles
• C. The area increases by a factor of 3
• D. The area increases by a factor of 4

Answer: (D)

When the radius of a circle is doubled, the effect on the area can be understood through the formula for the area of a circle, which is given by \( A = \pi r^2 \), where \( r \) represents the radius. Initially, let’s consider the area of the circle with the original radius \( r \). According to the formula, the area is \( A_1 = \pi r^2 \). Now, if the radius is increased to \( 2r \), the new area becomes: \[ A_2 = \pi (2r)^2 = \pi (4r^2) = 4\pi r^2 \] This shows that the new area \( A_2 \) is four times larger than the original area \( A_1 \). Thus, when the radius is doubled, the area increases by a factor of 4. This understanding clearly demonstrates that the correct answer is the one indicating that the area increases by a factor of 4, as it aligns directly with the mathematical relationship between the radius and area.

When the radius of a circle is doubled, the effect on the area can be understood through the formula for the area of a circle, which is given by ( A = \pi r^2 ), where ( r ) represents the radius.

Initially, let’s consider the area of the circle with the original radius ( r ). According to the formula, the area is ( A_1 = \pi r^2 ). Now, if the radius is increased to ( 2r ), the new area becomes:

[

A_2 = \pi (2r)^2 = \pi (4r^2) = 4\pi r^2

]

This shows that the new area ( A_2 ) is four times larger than the original area ( A_1 ). Thus, when the radius is doubled, the area increases by a factor of 4.

This understanding clearly demonstrates that the correct answer is the one indicating that the area increases by a factor of 4, as it aligns directly with the mathematical relationship between the radius and area.