What is the relationship between the radius and area of a circle?

The relationship between the radius and the area of a circle is defined by the formula ( A = \pi r^2 ), where ( A ) represents the area and ( r ) represents the radius. This formula clearly shows that the area is proportional to the square of the radius. As the radius increases, the area does not just increase at the same rate; rather, it increases by the square of that increase. For example, if the radius doubles, the area increases by a factor of four (since ( (2r)^2 = 4r^2 )). This is why the correct choice is that the area increases with the square of the radius.

In contrast, a linear increase would imply a direct one-to-one relationship, which is not the case with circles. An inverse relationship would mean that as the radius increases, the area would decrease, which contradicts the geometric understanding of how circles work. Additionally, stating that the area is constant regardless of changes in the radius would imply that any change in the radius has no effect on the area, which is untrue given that the area directly depends on the radius through squaring it. Thus, the area’s dependency on the square of the radius is what makes the