If a circle's diameter is doubled, how does the area change?

The area of a circle is calculated using the formula A = πr², where r represents the radius of the circle. When the diameter of the circle is doubled, the radius also changes, since the radius is half of the diameter.

If the original diameter is d, the original radius r would be d/2. When the diameter is doubled, the new diameter becomes 2d, resulting in a new radius of r' = 2d/2 = d. This means the new radius is twice as large as the original radius.

To find the area with the new radius, we substitute back into the area formula:

1. Original area (A) = π(r)² = π(d/2)² = π(d²/4)

2. New area (A') = π(r')² = π(d)² = π(d²)

Now, if we compare the new area to the original area, we have:

A' = π(d²) and A = π(d²/4).

A' = 4 × A.

Thus, when the diameter is doubled, the area of the circle becomes four times larger. This demonstrates why the correct answer indicates that the change in area is significant and multipl