What is the formula for the area of a circle?

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

To understand why this formula is effective, consider that the area of a shape measures the amount of space it occupies. For a circle, the area can be visualized as the sum of infinitely many infinitesimally small triangles formed from the center of the circle to its edge, culminating in the recognizable circular shape. The factor of π arises from the unique properties of circles, particularly the relationship between the circumference and the radius, which is integral in deriving this area formula.

The other options represent different geometric formulas and applications. The formula involving 2πr pertains to the circumference of the circle, while the term 4/3πr³ relates to the volume of a sphere. Lastly, the expression πr²h is used to calculate the volume of a cylinder, further demonstrating that each formula serves a distinct purpose in geometry. Thus, option A accurately captures the area calculation for circles specifically.