What is the formula for the area of a circle?

The formula for the area of a circle is given by πr², where r represents the radius of the circle. This equation is derived from integral calculus and reflects the idea that the area of a circle is proportional to the square of its radius.

When you square the radius, you account for the entire two-dimensional space within the circle. The constant π (pi) serves as a ratio that allows us to relate the circumference of the circle to its diameter and plays a crucial role in representing how much space the circle occupies.

In contrast, the other options do not accurately describe the area of a circle. The formula 2πr is known as the circumference of a circle, which measures the distance around the circle rather than the area within it. The formula r² simply represents the square of the radius, which lacks the necessary multiplier of π to define the area. Lastly, the formula πd also pertains to the circumference, where d is the diameter; again, it does not calculate the area. Thus, B is the correct choice because it correctly expresses the relationship between the radius and the area of a circle.