Which formula represents the area of a circle?

The area of a circle is defined by the formula A = πr², where A represents the area, r denotes the radius of the circle, and π (pi) is a constant approximately equal to 3.14159. This formula arises from the relationship between the radius and the space contained within the circle.

To better understand this formula, it can be visualized that as the radius increases, the area expands significantly because the area is proportional to the square of the radius. For instance, if you double the radius, the area increases by a factor of four (since (2r)² = 4r²). This is an important concept in geometry as it highlights how quickly area can grow compared to linear dimensions.

The other formulas presented do not accurately represent the area of a circle. For example, A = 2πr is actually the formula for the circumference of a circle, which represents the distance around the circle rather than the space enclosed by it. A = πr describes a concept related to circles but does not provide the correct measure of area, and A = r² does not incorporate the necessary π factor or adequately account for the circular shape. Thus, the formula A = πr² distinctly captures the relationship needed