Final answer:
The correct formula for the area of a circle is Area = πr^2 and can be rewritten in terms of the diameter as Area = π(½d)^2.
Step-by-step explanation:
The general formula for the area of a circle is Area = πr^2. This can be discerned by remembering that the area formula must relate to the dimensions of the circle, specifically, its radius (r) or diameter (d). Since the area is a two-dimensional measurement, it makes sense that the formula involves r squared, matching the dimensional units of square meters for area. In terms of the diameter, which is twice the radius (d = 2r), the formula can also be rewritten as Area = π(½d)^2.
Recalling that the perimeter (or circumference) of a circle and the area of a circle are different, we can distinguish that Area = 2πr refers to the circumference, not the area, and Area = 4πr implies a linear dimension times π, which does not give us a two-dimensional measurement; thus, these are incorrect for the area of a circle.