What is the formula for the area of a circle?

The formula for the area of a circle is derived from its geometric properties. The area of a circle is calculated using the radius, which is the distance from the center of the circle to any point on its circumference. The correct formula, A = πr², encapsulates this relationship where 'A' represents the area, 'π' (Pi) is a constant approximately equal to 3.14159, and 'r' stands for the radius of the circle.

In this context, the formula indicates that the area increases with the square of the radius. This is a fundamental principle in geometry that reflects how area scales with size—doubling the radius, for example, would result in quadrupling the area.

Examining the other options reveals that while they each pertain to circular measurements, they do not represent the area of the circle. The formula involving the circumference (2πr) measures the perimeter of the circle, while πd relates to the circumference as well but does not quantify area. Finally, the formula 1/2 x πr² is representative of the area of a sector of a circle, not the entire area of the circle itself. Thus, A = πr² is the precise expression for calculating the area