What is the formula for the area of a circle?

The area of a circle is calculated using the formula A = πr², where A represents the area and r is the radius of the circle. This formula arises from the relationship between the radius and the overall area of the circle. The radius is the distance from the center of the circle to any point on its circumference.

To understand why A = πr² accurately represents the area, consider that if you were to divide the circle into many small triangles or sectors, the area can be approximated by summing the areas of these shapes. As the number of divisions increases, this sum approaches the formula, and π serves as the crucial ratio that translates the radius into an area measurement, encapsulating the circular nature of the figure.

The other options provided do not represent the correct formula for the area of a circle. A = πr represents the circumference, A = 2πr is also related to the circumference, and A = πd uses the diameter (where d = 2r) but fails to yield the area directly without conversion. Thus, option B correctly provides the area measurement in relation to the radius squared, making it the authoritative formula for the area of a circle.