If the diameter of a circle is doubled, what happens to the area?

If the diameter of a circle is doubled, what happens to the area?

• A. The area remains the same
• B. The area is doubled
• C. The area is quadrupled
• D. The area decreases by half

Answer: (C)

When the diameter of a circle is doubled, the radius also doubles. The area of a circle is calculated using the formula \(A = \pi r^2\), where \(r\) represents the radius. If the original radius is \(r\), then doubling the diameter results in a new radius of \(2r\). Substituting this new radius into the area formula gives: \[ A = \pi (2r)^2 \] This simplifies to: \[ A = \pi \cdot 4r^2 = 4(\pi r^2) \] This shows that the new area is four times the original area. Therefore, when the diameter is doubled, the area of the circle is indeed quadrupled. Understanding this relationship between the diameter, radius, and area is crucial in geometry, particularly in solving problems related to circles.

When the diameter of a circle is doubled, the radius also doubles. The area of a circle is calculated using the formula (A = \pi r^2), where (r) represents the radius.

If the original radius is (r), then doubling the diameter results in a new radius of (2r). Substituting this new radius into the area formula gives:

[

A = \pi (2r)^2

]

This simplifies to:

[

A = \pi \cdot 4r^2 = 4(\pi r^2)

]

This shows that the new area is four times the original area. Therefore, when the diameter is doubled, the area of the circle is indeed quadrupled. Understanding this relationship between the diameter, radius, and area is crucial in geometry, particularly in solving problems related to circles.