How is the area of a circle represented mathematically?

The area of a circle is mathematically represented as A = πr², where A represents the area, π is a constant approximately equal to 3.14159, and r denotes the radius of the circle. This formula arises from integrating the infinitesimal contributions of the circular shape, which can be derived from the relationship between the radius and the overall dimensions of the circle.

The rationale behind this formula is that the area is proportional to the square of the radius because, as the radius increases, the space covered within the circle increases exponentially. Each infinitesimal portion of the circle's area relates to a small increase in radius, and thus the area expands by a factor related to the square of that radius.

On the other hand, the other options represent different concepts. For instance, A = 2πr represents the circumference of a circle, which is the distance around the circle rather than the area contained within it. The formula A = πd uses the diameter (which is double the radius), and while it can also yield the area, it is less commonly used than the radius-based formula. Lastly, A = 4πr² is not a formula for the area of a circle; instead, it represents the surface