What is the formula to calculate the area of a circle?

The formula to calculate the area of a circle is derived from the relationship of the circle's radius to its area. Specifically, the area is defined as the amount of space enclosed within a circle. The formula ( A = \pi r² ) effectively indicates that the area ( A ) is equal to pi (π) multiplied by the radius (r) squared.

This squared relationship arises from the nature of geometry, where the radius represents the distance from the center of the circle to any point on its boundary, and pi is a constant that relates the diameter of a circle to its circumference. As a result, this formula not only accurately reflects the mathematical properties of circles but also aligns with empirical observations, making it foundational in geometry and relevant in many applications, such as engineering and physics.

In contrast, the other choices either represent different formulas pertaining to aspects of circles (like circumference) or are not relevant to area at all. The formula for circumference, for example, uses ( A = 2\pi r ) or a similar relation involving the diameter, showing that a solid understanding of each formula's application helps differentiate between them.